Solved: Volume of a Balloon The volume V (in cubic meters)

Find f(3) if f(x) = -4x2 + 5x. (pp. 212-214)

Find f(3x) If f(x) = 4 - 2x2. (pp. 212-214)

Find the domain of the function f(x) = x2 - 1 (pp. 215-216)

If f(x) = x + 1 and g(x) = x3, then __ = (x + 1)3

True or False f(g(x = f(x) g(x).

True or False The domain of the composite function (f 0 g) (x) is the same as the domain of g(x).

In Problems 7 and 8, evaluate each expression using the values given in the table. , x -3 -2 -1 0 2 3 f(x) -7 -5 -3 -1 3 5 5 g(x) 8 3 0 -1 0 3 8 (a) (f 0 g)(l) (b) (f 0 g) ( -1 ) (c) (g 0 f)( -1) (d) (g 0 f)(0) (e) (g 0 g)( -2) (f) (f 0 f)( -1)

In Problems 7 and 8, evaluate each expression using the values given in the table. x -3 - 2 -1 0 2 3 f(x) 11 9 7 5 3 -1 g(x) -8 -3 0 0 -3 -8 (a) (f 0 g)(l) (b) (f 0 g)(2) (c) (g 0 f)(2) (d) (g 0 f) (3) (e) (g 0 g)(l) (f) (f 0 f) (3)

In Problems 9 and 10, evaluate each expression using the graphs of y = f(x) and y = g(x) shown below. y y=g(x) (6,5) (7,5) A (1,4) (5,4)r .. (8,4) (-1,3) (-1,1 ) I I 1 '7" /, (5'11 ) 1 74 I II -2 I'\.. 2 6 8 x -2 \ y --------y = f(x) (2, -2) (1, -1) (a) g(f( - 1 (b) g(f(O (c) f(g( - 1 (d) f(g(4

In Problems 9 and 10, evaluate each expression using the graphs of y = f(x) and y = g(x) shown below. y y=g(x) (6,5) (7,5) A (1,4) (5,4)r .. (8,4) (-1,3) (-1,1 ) I I 1 '7" /, (5'11 ) 1 74 I II -2 I'\.. 2 6 8 x -2 \ y --------y = f(x) (2, -2) (1, -1) (a) g(f(l (b) g(f(5 (c) f(g(O (d) f(g(2

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = 2x; g(x) = 3x2 + 1

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = 3x + 2; g(x) = 2x2 - 1

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = 4x - 3; g(x) = 3 - 2 x

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = 2x2; g(x) = 1 - 3x2

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = v'X; g(x) = 2x

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = -Vx+l; g(x) = 3x

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = Ixl ; g(x) = x2 + 1

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = Ix - 21; g(x) = x2 + 2

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = --; g(x) = x

In Problems 11-20, for the given functions f and g, find: (a) (f 0 g) (4) (b) (g 0 f)(2) (c) (f 0 f)(1) (d) (g 0 g)(O) f(x) = x3/2; g (x) = X + 1

In Problems 21-28, find the domain of the composite function f o g. f(x) = 3 _; g(x) = x -I x

In Problems 21-28, find the domain of the composite function f o g. f(x) = --; g(x) = -- x + 3 x

In Problems 21-28, find the domain of the composite function f o g. f ( x) = x-I ; g ( x) = -

In Problems 21-28, find the domain of the composite function f o g. f(x) = x_; g(x) = x + 3 x

In Problems 21-28, find the domain of the composite function f o g. f(x) = Vx; g(x) = 2x + 3

In Problems 21-28, find the domain of the composite function f o g. f(x) = x - 2; g(x) =

In Problems 21-28, find the domain of the composite function f o g. f(x) = xl + 1; g(x) =

In Problems 21-28, find the domain of the composite function f o g. f(x) = Xl + 4; g(x) =

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = 2x + 3; g(x) = 3x

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = -x; g(x) = 2x - 4

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = 3x + 1; g(x) = Xl

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x + 1; g(x) = x2 + 4

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = xl; g(x) = x2 + 4

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x2 + 1; g(x) = 2x2 + 3

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x-I ; g(x) =

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = --; g(x) = -- x+3 x

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x -I ; g(x) =-

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x + 3 ; g(x) =

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = Vx; g(x) = 2x + 3

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) =; g(x) = 1 - 2x

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x2 + 1; g(x) =

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x2 + 4; g(x) =

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = --x+1 ; g(x) =--x - 3

In Problems 29-44, for the given functions f and g, find: (a) fog (b) g 0 f (c) f 0 f (d) gog State the domain of each composite function. f(x) = x _ 2 ; g(x) = 2x - 5

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = 2x; g(x) = 2 "x

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = 4x; g(x) = 4 x

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = x3; g(x) = -

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = x + 5; g(x) = x - 5

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = 2x - 6; g(x) = 2 "(x + 6)

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = 4 - 3x; g(x) = 3 (4 - x)

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = ax + b; g(x) = -(x - b) a"* 0

In Problems 45-52, show that (f 0 g)(x) = (g 0 f)(x) = x. f(x) = -; g(x) = - x x

In Problems 53-58, find functions f and g so that fog = H. H(x) = (2x + 3) 3 4

In Problems 53-58, find functions f and g so that fog = H. H(x) = (1 + x2)

In Problems 53-58, find functions f and g so that fog = H. H(x) =

In Problems 53-58, find functions f and g so that fog = H. H(x) =

In Problems 53-58, find functions f and g so that fog = H. H(x) = 12x + 11

In Problems 53-58, find functions f and g so that fog = H. H(x) = 12x2 + 31

If f(x) = 2x3 - 3x2 + 4x - 1 and g(x) = 2, find (f 0 g )(x) and (g 0 f)(x).

If f(x) = --, find (f 0 f)(x). x-I

11 f(x) = 2x2 + 5 and g(x) = 3x + a, find a so that the graph of fog crosses the y-axis at 23.

If f(x) = 3x2 - 7 and g(x) = 2x + a, find a so that the graph of fog crosses the y-axis at 68.

In Problems 63 and 64, use the functions f and g to find: (a) fog (b) g 0 f c) the domain of fog and of g 0 f (d) the conditions for which fog = g 0 f f(x) = ax + b; g(x) = cx + d

In Problems 63 and 64, use the functions f and g to find: (a) fog (b) g 0 f c) the domain of fog and of g 0 f (d) the conditions for which fog = g 0 f f(x) = --; g(x) = mx cx + d

Surface Area of a Balloon The surface area S (in square meters) of a hot-air balloon is given by S(r) = 47Tr2 where r is the radius of the balloon (in meters). If the radius r is increasing with time t (in seconds) according to the 2 , . formula r(t) = to, t ?: 0, fmd the surface area S of the bal.) loon as a function of the time t

Volume of a Balloon The volume V (in cubic meters) of the hot-air balloon described in Problem 65 is given by V(r) = 7Tr3. If the radius r is the same function of t as in Problem 65, find the volume Vas a function of the time t.

Automobile Production The number N of cars produced at a certain factory in 1 day after t hours of operation is given byN(t) = l OOt - 5t2, 0::; t::; 10. If the cost C (in dollars) of producing N cars is C(N) = 15,000 + S OOON, find the cost C as a function of the time t of operation of the factory.

Environmental Concerns The spread of oil leaking from a tanker is in the shape of a circle. If the radius r (in feet) of the spread after t hours is r( t) = 200Vt, find the area A of the oil slick as a function of the time t.

Production Cost The price p, in dollars, of a certain product and the quantity x sold obey the demand equation 1 p = --x + 100 0::; x ::; 400 4 Suppose that the cost C, in dollars, of producing x units is Vx C = 25 + 600 Assuming that all items produced are sold, find the cost Cas a function of the price p. [Hint: Solve for x in the demand equation and then form the composite.]

Cost of a Commodity The price p, in dollars, of a certain commodity and the quantity x sold obey the demand equation 1 p = -"5 x + 200 0::; x ::; 1000 Suppose that the cost C, in dollars, of producing x units is Yx C = 10 + 400 Assuming that all items produced are sold, find the cost Cas a function of the price p.

Volume of a Cylinder The volume V of a right circular cylinder of height h and radius r is V = 7Tr2h. If the height is twice the radius, express the volume Vas a function of I:

Volume of a Cone The volume V of a right circular cone is V = 7Tr2h. If the height is twice the radius, express the vol.) ume V as a function of r.

Foreign Exchange Traders often buy foreign currency in hope of making money when the currency's value changes. For example, on February 17, 2006, one U.S. dollar could purchase 0.8382 Euros, and one Euro could purchase 140.9687 yen. Let f(x) represent the number of Euros you can buy with x dollars, and let g( x) represent the number of yen you can buy with x Euros. (a) Find a function that relates dollars to Euros. (b) Find a function that relates Euros to yen. (c) Use the results of parts (a) and (b) to find a function that relates dollars to yen. That is, find (g 0 f) (x) = g(f(x)). (d) What is g(f(1000))?

Temperature Conversion The function C(F) = "9(F - 32) converts a temperature in degrees Fahrenheit, F, to a temperature in degrees Celsius, C. The function K( C) = C + 273, converts a temperature in degrees Celsius to a temperature in kelvins, K. (a) Find a function that converts a temperature in degrees Fahrenheit to a temperature in kelvins. (b) Determine 80 degrees Fahrenheit in kelvins.

If f and g are odd functions, show that the composite function fog is also odd.

If f is an odd function and g is an even function, show that the composite functions f og and g 0 f are both even.

Is the set of ordered pairs {(I, 3), ( 2,3), (-1,2)}a function? Why or why not? (pp. 208-210)

Where is the function f(x) = x 2 increasing? Where is it decreasing? (pp. 233-234)

What is the domain of f ( x) = 2 ?(pp. 215-216) x + 3x - 18

If every horizontal line intersects the graph of a function f at no more than one point, f is a(n) __ function.

If rl denotes the inverse of a function f, then the graphs of f and rl are symmetric with respect to the line ___.

If the domain of a one-to-one function f is [4, 00 ) , the range of its inverse, f - I , is ___ .

True or False If f and g are inverse functions, the domain of f is the same as the domain of g.

True or False If f and g are inverse functions, their graphs are symmetric with respect to the line y = x.

In Problems 9-16, determine whether the function is one-to-one. Domain 20 Hou rs 25 Hou rs 30 Hours 40 Hou rs Range $200 $300 $350 $425

In Problems 9-16, determine whether the function is one-to-one. Domain Bob Dave John Chuck Range t--- Karla Debra Dawn Phoebe

In Problems 9-16, determine whether the function is one-to-one. Domain 20 Hours 25 Hours - 30 Hours 40 Hours Range ...j.- $200 $350 $425

In Problems 9-16, determine whether the function is one-to-one. Domain Bob Dave John Chuck Range Karla Debra ....j.- Phoebe

In Problems 9-16, determine whether the function is one-to-one. {(2,6), (-3,6), ( 4,9), ( 1,10)}

In Problems 9-16, determine whether the function is one-to-one. {( - 2,5), (-1,3), (3,7), ( 4, 12)}

In Problems 9-16, determine whether the function is one-to-one. {CO, 0), ( 1, 1), ( 2, 16), (3, 81)}

In Problems 9-16, determine whether the function is one-to-one. {(1, 2), ( 2,8), (3, 18), ( 4,32)}

In Problems 17-22, the graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. -3 3 x

In Problems 17-22, the graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. y 3-3 3 x

In Problems 17-22, the graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. -3 3 x -3

In Problems 17-22, the graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. Y x

In Problems 17-22, the graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. -3

In Problems 17-22, the graph of a function f is given. Use the horizontal-line test to determine whether f is one-to-one. -3 Y 3 3 x

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. Location (inches) Mt Waialeale, Hawaii Monrovia, Liberia Pago Pago, American Samoa Moulmein, Bu rma Lae, Papua New Guinea Source: Information Please Almanac 460.00 202 01 196.46 191.02 182.87

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. Title (in millions) Star Wars $461 Star Wars: Episode One - The $431 Phantom Menace E. T. the Extra Terrestrial $400 Jurassic Park $357 Forrest Gump $330

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. Age 30 40 45 Source: eterm.com 460.00 202 01 196.46 191.02 182.87 Monthly Cost of Life Insurance $7.09 $8.40 $11.

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. State Unemployment Rate Virginia Nevada Tennessee Texas Source: United States Statistical Abstract

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. {( -3, 5), (-2, 9), (-1, 2), (0, ll), (1, -5)}

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. {( -2, 2), (-1,6), (0, 8), (1, -3), (2, 9)}

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. { (-2, 1), (-3,2), (-10,0) , (1, 9), (2, 4)}

In Problems 23-30, find the inverse of each one-to-one function. State the domain and the range of each inverse function. {( -2, -8), ( -1, -1), (0,0), (1, 1), (2, 8)}

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = 3x + 4; g (x) = 3 (x - 4)

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = 3 - 2x; g (x) = _ l(x - 3) 2

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = 4x - 8; g (x) = "4 + 2

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = 2x + 6; g (x) = -x - 3 2

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f(x) = x3 - 8; g (x) =

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = (x - 2j2, x 2: 2; g (x) = vX + 2

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = -; g (x) = - x x

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = x; g (x) = x

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = --x+4 ; g (x) = --2-x

In Problems 31-40, verify that the functions f and g are inverses of each other by showing that f (g (x)) = x and g (f (x)) = x. Give any values of x that need to be excluded. f (x) = -- 2x + 3 ; g (x) = - 1 -- 2x

In Problems 41-46, the graph of a one-to-one function f is given. Draw the graph of the inverse function rl. For convenience (and as a hint), the graph of y = x is also given. Y 3 -3 Y= x (1,2) 3 x

In Problems 41-46, the graph of a one-to-one function f is given. Draw the graph of the inverse function rl. For convenience (and as a hint), the graph of y = x is also given. Y 3 -3 y=X 3 x

In Problems 41-46, the graph of a one-to-one function f is given. Draw the graph of the inverse function rl. For convenience (and as a hint), the graph of y = x is also given. y Y=X 3 (2,1) -3 3 x ( -1, -1) -3

In Problems 41-46, the graph of a one-to-one function f is given. Draw the graph of the inverse function rl. For convenience (and as a hint), the graph of y = x is also given. -3 Y 3 -3 Y=X 3 x (1 , -1)

In Problems 41-46, the graph of a one-to-one function f is given. Draw the graph of the inverse function rl. For convenience (and as a hint), the graph of y = x is also given. 3 -3 Y=X 3 x

In Problems 41-46, the graph of a one-to-one function f is given. Draw the graph of the inverse function rl. For convenience (and as a hint), the graph of y = x is also given. 2 -2 Y=X 2 x

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) = 3x

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) = -4x

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) = 4x + 2

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) = 1 - 3x

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) = x3 - 1

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [( x) = x3 + 1

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [( x) = x2 + 4 xo

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [( x) = x2 + 9 xo

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) = - x

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) = - - x

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) =--x-2

In Problems 47-58, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain and the range o[ [ and rl . Graph [, [-I, and y = x on the same coordinate axes. [(x) =--x+2

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = --3 + x

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = -- 2 - x

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) =-x+2

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = - - x-I

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = - .)x - 1

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = ---x

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = --2x - 3

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [( x) = 2x - 3 x+4

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = 2x + 3 x + 2

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = -3x - 4 x - 2

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = x2 - , 4 x>

In Problems 59-70, the [unction [ is one-to-one. Find its inverse and check your answeJ: State the domain o[ [ and find its range using rl . [(x) = --, x> 3x

Use the graph of y = [(x) given in Problem 41 to eval uate the following: (a) [( -1) (b) [(1) (c) rl (l) (d) rl(2)

Use the graph of y = [(x) given in Problem 42 to evaluate the following: (a) [(2) (b) [(1) (c) rl (O) (d) rl (- l)

If [(7) = 13 and [ is one-to-one, what is rl (B)?

If g (-5) = 3 and g is one-to-one,what is g-I(3)?

The domain of a one-to-one function [is [5, 00), and its range is [-2, 00). State the domain and the range of rl .

The domain of a one-to-one function [is [0, 00), and its range is [5, 00). State the domain and the range of rl.

The domain of a one-to-one function g is the set of all real numbers, and its range is [0,00). State the domain and the range of g-I .

The domain of a one-to-one function g is [0, 15], and its range is (0, 8). State the domain and the range of g-I.

A function y = [(x) is increasing on the interval (0, 5). What conclusions can you draw about the graph of y = rl(x)?

A function y = [(x) is decreasing on the interval (0,5). What conclusions can you draw about the graph of y = rl(x)?

Find the inverse of the linear function [(x) = mx + b In "*

Find the inverse of the function [(x) = Vr2 - x2 O:=; X :=; r

A function [ has an inverse function. If the graph of [lies in quadrant I, in which quadrant does the graph of r1 lie?

A function [ has an inverse function. If the graph of [lies in quadrant II, in which quadrant does the graph of r1 lie?

The function [(x) = Ixl is not one-to-one. Find a suitable restriction on the domain of [so that the new function that results is one-to-one. Then find the inverse off.

The function [(x) = x4 is not one-to-one. Find a suitable restriction on the domain of [ so that the new function that results is one-to-one. Then find the inverse of f.

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f(x) to represent a function, an applied problem might use C = C(q) to represent the cost C of manufacturing q units of a good since, in economics, q is used for output. Because of this, the inverse notation f-1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rathel; the inverse of a function such as C = C( q) will be q = q( C). So C = C( q) is a function that represents the cost C as a function of the output q, while q = q( C) is a jil.l1ction that represents the output q as a function of the cost C. Problems 87-90 illustrate this idea. Vehicle Stopping Distance Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function d(r) = 6.97r - 90.39 (a) Express the speed r at which the car is traveling as a function of the distance d required to come to a complete stop. (b) Verify that r = red) is the inverse of d = d(r ) by showing that r(d(r) ) = r and d(r(d)) = d. (c) Predict the speed that a car was traveling if the distance required to stop was 300 feet.

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f(x) to represent a function, an applied problem might use C = C(q) to represent the cost C of manufacturing q units of a good since, in economics, q is used for output. Because of this, the inverse notation f-1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rathel; the inverse of a function such as C = C( q) will be q = q( C). So C = C( q) is a function that represents the cost C as a function of the output q, while q = q( C) is a jil.l1ction that represents the output q as a function of the cost C. Problems 87-90 illustrate this idea. Height and Head Circumference The head circumference C of a child is related to the height H of the child (both in inches) through the function H (C) = 2.15C - 1 0.53 (a) Express the head circumference C as a function of height H. (b) Verify that C = C(H) is the inverse of H = H(C) by showing that H(C(H)) = H and C(H(C)) = C. (c) Predict the head circumference of a child who is 26 inches tall.

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f(x) to represent a function, an applied problem might use C = C(q) to represent the cost C of manufacturing q units of a good since, in economics, q is used for output. Because of this, the inverse notation f-1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rathel; the inverse of a function such as C = C( q) will be q = q( C). So C = C( q) is a function that represents the cost C as a function of the output q, while q = q( C) is a jil.l1ction that represents the output q as a function of the cost C. Problems 87-90 illustrate this idea. Ideal Body Weight The ideal body weight W for men (in kilograms) as a function of height h (in inches) is given by the function W(h) = SO + 2.3(h - 60) (a) What is the ideal weight of a 6-foot male? (b) Express the height h as a function of weight W. (c) Verify that h = heW) is the inverse of W = W(h) by showing that h(W(h)) = h and W(h(W)) = W. (d) What is the height of a male who is at his ideal weight of 80 kilograms? Note: The ideal body weight W for women (in kilograms) as a function of height h (in inches) is given by W(h) = 45.5 + 2.3(h - 60).

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f(x) to represent a function, an applied problem might use C = C(q) to represent the cost C of manufacturing q units of a good since, in economics, q is used for output. Because of this, the inverse notation f-1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rathel; the inverse of a function such as C = C( q) will be q = q( C). So C = C( q) is a function that represents the cost C as a function of the output q, while q = q( C) is a jil.l1ction that represents the output q as a function of the cost C. Problems 87-90 illustrate this idea. Temperature Con"ersion The function F(C) = s C + 32 converts a temperature from C degrees Celsius to F degrees Fahrenheit. (a) Express the temperature in degrees Celsius C as a function of the temperature in degrees Fahrenheit F. (b) Verify that C = C(F) is the inverse of F = F(C) by showing that C(F(C) ) = C and F(C(F)) = F. (c) What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?

Income Taxes The function T(g) = 4220 + 0.25(g - 30,650) represents the 2006 federal income tax T (in dollars) due for a "single" filer whose adjusted gross income is g dollars, where 30,650 g 74,200. (a) What is the domain of the function T? (b) Given that the tax due T is an increasing linear function of adjusted gross income g, find the range of the function T. (c) Find adjusted gross income g as a function of federal income tax T. What are the domain and the range of this function?

Income Taxes The function T(g) = 1510 + 0.15(g - 15,100) represents the 2006 federal income tax T (in dollars) due for a "married filing jointly" filer whose adjusted gross income is g dollars, where 15,100 g 61,300. (a) What is the domain of the function T? (b) Given that the tax due T is an increasing linear function of adjusted gross income g, find the range of the function T. (c) Find adjusted gross income g as a function of federal income Lax T. What are the domain and the range of this function?

Gra"ity on Earth If a rock falls from a height of 100 meters on Earth, the height H (in meters) after t seconds is approximately H(t) = 100 - 4.9t2 (a) In general, quadratic functions are not one-to-one. However, the function H(t) is one-to-one. Why? (b) Find the inverse of H and verify your result. ( c) How long will it take a rock to fall 80 meters?

Period of a Pendululll The period T (in seconds) of a simple pendulum as a function of its length I (in feet) is given by T(l) = 27T I 32.2 (a) Express the length I as a function of the period T. (b) How long is a pendulum whose period is 3 seconds?

Given f(x) = ax + b cx + d find r1 (x). If c oF 0, under what conditions on a, b, c, and d is f = rl?

Can a one-to-one function and its inverse be equal? What must be true about the graph of f for this to happen? Give some examples to support your conclusion.

Draw the graph of a one-to-one function that contains the points ( -2, -3 ), (0, 0), and (1, 5) . Now draw the graph of its inverse. Compare your graph to those of other students. Discuss any similarities. What differences do you see?

Give an example of a function whose domain is the set of real numbers and that is neither increasing nor decreasing on its domain, but is one-to-one. [Hint: Use a piecewise-defined function.]

Is every odd function one-to-one? Explain.

Suppose C(g) represents the cost C, in dollars, of manufacturing g cars. Explain what C-1 (800,000) represents.

4 3 = __ ; 8 2/3 = __ ; T2 = __ . (pp. 21-24 and pp. 75-76)

Solve: 5x - 2 = 3 (pp. 86-92)

True or False To graph y = (x - 2)3 , shift the graph of y = x 3 to the left 2 units. (pp. 252-260)

Find the average rate of change of f ( x) = 3x - 5 from pp. 75-76) x = 0 to x = 4. (pp. 236-238)

True or False The function f ( x) = -- has y = 2 as a horizontal asymptote. (pp. 346-352) x - j

The graph of every exponential function f(x ) = aX, a > 0, a =ft 1, passes through three points: __ , __ , and ___.

If the graph of the exponential function f(x) = aX, a > 0, a =ft 1, is decreasing, then a must be less than __ .

lf 3x = 3 4 , then x = __ .

True or False The graphs of y = Y and y = - are identical. 3

True or False The range of the exponential function f(x) = a'r, a > 0, a =ft 1, is the set of all real numbers.

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. (a) 3 2 . 2 (b) 3 223 (c) 3 2236 (d) 3 0

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. a) 5 17 (b) 5 173 (c) 5 1732 (d) 50

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. (a) 23.14 . 7 (b) 23 141 (c) 23.1415 (d) 2'

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. a) 22.7 . 85 (b) 22.71 (d) 2e

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. (a) 3.12 . 7 (b) 3.142 71 (c) 3.1412 718 (d) 7re

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. a) 2.73 1 b) 2.71 3 14 (c) 2.7183.141 (d) er

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. e 1.2

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. e -1.3

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. e-O . 85

In Problems 11-20, approximate each number using a calculator. Express your answer rounded to three decimal places. e2.1

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x x x x F(x) -1 3 0 6 12 2 18 3 30

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x x x F(x) -1 2 0 5 8 2 11 3 14

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x x F(x) -1 0 4 4 2 16 3 64

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x F(x) -1 3 2 1 0 2 4 27 3 8

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x x x x F(x) -1 0 3 6 2 12 3 24

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x x x F(x) -1 6 0 0 2 3 3 10

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x x F(x) -1 2 0 4 6 2 8 3 10

In Problems 21-28, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base. [Hint: Look at the ratio of consecutive values.] x F(x) -1 -1 2 0 4 8 2 16 3 32 2 16 3 32

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y -2 -1

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y -2 -1

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y -2 2X

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y 2X

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y 2 2x -1

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y y 3 ) y = o -2

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y 2x

In Problems 29-36, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = 3x B. Y = TX C. y = -3 x D. y = -r' E. y = 3x - 1 F. Y = 3"- 1 G. y = 3J -x H. Y = 1 - Y O 2x -2 2x -1

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 2.1 + 1

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = Y - 2

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 3x- 1

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 2 x+2

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 3 or

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 4 Gr

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = r' - 2

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = -3.1 + 1

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 2 + 4.1-1

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 1 - 2.1+3

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 2 + 3xI 2

In Problems 37-48, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 1 - Txl3

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = e-X

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = -ex

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = ex+2

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f ( x) = eX - 1

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 5 - e-X

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 9 - 3e-x

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 2 - e-xl 2

In Problems 49-56, begin with the graph of y = eX [Figure 27J and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. f(x) = 7 - 3e2x

In Problems 57-76, solve each equation. 7 x = 73

In Problems 57-76, solve each equation. 5.1 = 5-6

In Problems 57-76, solve each equation. TX = 16

In Problems 57-76, solve each equation. TX = 81

In Problems 57-76, solve each equation. Gr 1 25

In Problems 57-76, solve each equation. l)X 1 - = - 4 64

In Problems 57-76, solve each equation. 22x -1 = 4

In Problems 57-76, solve each equation. 5x+3 1 = - 5

In Problems 57-76, solve each equation. 3x3 = 9x

In Problems 57-76, solve each equation. 4.12 = 2.1

In Problems 57-76, solve each equation. 8-.1+14 = 1 6.1

In Problems 57-76, solve each equation. 9-x+ 15 = 27.1

In Problems 57-76, solve each equation. y.2_7 = 272.1

In Problems 57-76, solve each equation. 5.1' +8 = 1252.1

In Problems 57-76, solve each equation. 4.1 2x2 = 1 62

In Problems 57-76, solve each equation. 92x 27x' = T1

In Problems 57-76, solve each equation. eX = e 3 x+S

In Problems 57-76, solve each equation. e3x = e2-x

In Problems 57-76, solve each equation. eX = e .r "1 e

In Problems 57-76, solve each equation. (e4y . e X ' = e l2

Suppose that f(x) = 2'. (l)X 1 62. - = - 4 64 66. 4.12 = 2.1 70. 5.1' +8 = 1252.1 (a) What is f(4) ? What point is on the graph of f? (b) If f( x) = 1 1 6 , what is x? What point is on the graph of f?

Suppose that f(x) = 3x. (a) What is f( 4)? What point is on the graph of f? (b) If f(x) = %, what is x? What point is on the graph of f?

Suppose that g(x) = 4.1 + 2. (a) What is g( -I)? What point is on the graph of g? (b) If g(x) = 66, what is x? What point is on the graph of g?

Suppose that g(x) = 5x - 3. (a) What is g( -I)? What point is on the graph of g? (b) If g(x) = 122, what is x? What point is on the graph of g?

Suppose that H(x) = 3 ' "2 - 2. (a) What is H( -2) ? What point is on the graph of H? (b) If H(x) = _ 1 :, what is x? What point is on the graph of H?

Suppose that F(x) = -2 :3 + 1. (a) What is F ( - I ) ? What point is on the graph of F? (b) If F(x) = -53, what is x? What point is on the graph of F?

If 4.1 = 7, what does 4-2.1 equal?

If 2.1 = 3, what does 4-x equal?

If TX = 2, what does 32.1 equal?

If 5-x = 3, what does 53.1 equal?

In Problems 87-90, determine the exponential function whose graph is given. y 88. y 20 16 1 2 (-1 , ) y = O -3 -2 -1 2 3 x

In Problems 87-90, determine the exponential function whose graph is given. y 20 16 1 2 8 (-1 , ) -3 -2 -1 -2 -1 2 3 x y = O

In Problems 87-90, determine the exponential function whose graph is given. Y (0, -1 ) (-1 , - ) 1-7---:3X Y = 0 -10 -20 -30 (2, -36) -40

In Problems 87-90, determine the exponential function whose graph is given. y 2 / (-1 , -) -4 -8 -12 (0, -1 ) y = o 3

In Problems 91-94, graph each function. Based on the graph, state the domain and the range and find any intercepts. { e-X if x < 0 { eX if x < 0

In Problems 91-94, graph each function. Based on the graph, state the domain and the range and find any intercepts. f(x) = -r e if x 0 e' if x 0

In Problems 91-94, graph each function. Based on the graph, state the domain and the range and find any intercepts. { -eX f(x) = -e - x if x < if x

In Problems 91-94, graph each function. Based on the graph, state the domain and the range and find any intercepts. f(x) = {=::x if x < 0 if x

Optics If a single pane of glass obliterates 3% of the light passing through it, the percent p of light that passes through n successive panes is given approximately by the function pen) = 100(0.97)" (a) What percent of light will pass through 10 panes? (b) What percent of light will pass through 25 panes?

Atmospheric Pressure The atmospheric pressure p on a balloon or plane decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height h (in kilometers) above sea level by the function p(h) = 760e-0 14S h (a) Find the atmospheric pressure at a height of 2 kilometers (over a mile). (b) What is it at a height of 10 kilometers (over 30,000 feet)?

Depreciation The price p, in dollars, of a Honda Civic DX Sedan that is x years old is given by p(x) = 16,630(0.90Y (a) How much does a 3-year-old Civic DX Sedan cost? (b) How much does a 9-year-old Civic DX Sedan cost?

Healing of Wounds The normal healing of wounds can be modeled by an exponential function. If Ao represents the original area of the wound and if A equals the area of the wound, then the function A(n) = Aoe-O . 3Sn describes the area of a wound after n days following an injury when no infection is present to retard the healing. Suppose that a wound initially had an area of 100 square millimeters. (a) If healing is taking place, how large will the area of the wound be after 3 days? (b) How large will it be after 10 days?

Drug Medication The function D(h) = 5e-OAh can be used to find the number of milligrams D of a certain drug that is in a patient's bloodstream h hours after the drug has been administered. How many milligrams will be present after 1 hour? After 6 hours?

Spreading of Rumors A model for the number N of people in a college community who have heard a certain rumor is N = P(l - e-O.15d) where P is the total population of the community and d is the number of days that have elapsed since the rumor began. In a community of 1000 students, how many students will have heard the rumor after 3 days?

Exponential Probability Between 12:00 PM and 1:00 PM, cars arrive at Citibank's drive-thru at the rate of 6 cars per hour (0.1 car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 12:00 PM: F(t) = 1 - e-011 (a) Determine the probability that a car will arrive within 10 minutes of 12:00 PM (that is, before 12:10 PM). (b) Determine the probability that a car will arrive within 40 minutes of 12:00 PM (before 12:40 PM). (c) What value does F approach as t becomes unbounded in the positive direction? 6:l (d) Graph F using a graphing utility. (e) Using TRACE, determine how many minutes are needed for the probability to reach 50%.

Exponential Probability Between 5:00 PM and 6:00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour (0.15 car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 5:00 PM: F(t) = 1 - e-0 15 1 (a) Determine the probability that a car will arrive within 15 minutes of 5:00 PM (that is, before 5:15 PM). (b) Determine the probability that a car will arrive within 30 minutes of 5:00 PM (before 5:30 PM). (c) What value does F approach as t becomes unbounded in the positive direction? (e1 ) Graph F using a graphing utility. (e ) Using TRACE, determine how many minutes are needed for the probability to reach 60%.

Poisson Probability Between 5:00 PM and 6:00 PM, cars arrive at McDonald's drive-thru at the rate of 20 cars per hour. The following formula from probability can be used to determine the probability that x cars will arrive between 5:00 PM and 6:00 PM. P(x) where x! = x'(x - 1 )'(x - 2) .. 3 2 1 (a) Determine the probability that x = 15 cars will arrive between 5:00 PM and 6:00 PM. (b) Determine the probability that x = 20 cars will arrive between 5:00 PM and 6:00 PM.

Poisson Probabilit), People enter a line for the Demon Roller Coaster at the rate of 4 per minute. The following formula from probability can be used to determine the probability that x people will arrive within the next minute. where 4xe-4 P(x) = ,x. x! = x ' ( x - 1 ) ' ( x - 2) .. 3 2 1 (a) Determine the probability that x = 5 people will arrive within the next minute. (b) Determine the probability that x = 8 people will arrive within the next minute.

Relative Humidity The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that it can hold at a specific temperature. The relative humidity, R, is found using the following formula: ( 4221 4221 + ) R = 10 T+459.4 -D+459.4 2 where T is the air temperature (in OF) and D is the dew point temperature (in OF). (a) Determine the relative humidity if the air temperature is 50 Fahrenheit and the dew point temperature is 41 Fahrenheit. (b) Determine the relative humidity if the air temperature is 68 Fahrenheit and the dew point temperature is 59 Fahrenheit. (c) What is the relative humidity if the air temperature and the dew point temperature are the same?

Learning Curve Suppose that a student has 500 vocabulary words to learn. If the student learns 15 words after 5 minutes, the function L(t) = 500(1 - e-0 006 1 t ) approximates the number of words L that the student will learn after t minutes. (a) How many words will the student learn after 30 minutes? (b) How many words will the student learn after 60 minutes?

Current in a RL Circuit The equation governing the amount of current 1 (in amperes) after time / (in seconds) in a single RL circuit consisting of a resistance R (in ohms), an inductance L (in henrys), and an electromotive force E (in volts) is 1 = (1 - e-(R/L)t ] R E [ (a) If E = 120 volts, R = 10 ohms, and L = 5 henrys, how much current I, is flowing after 0.3 second? After 0.5 second? After 1 second? (b) What is the maximum current? (c) Graph this function 1 = 1] (/), measuring 1 along the y-axis and / along the x-axis. (d) If E = 1 20 volts, R = 5 ohms, and L = 10 henrys, how much current 1 2 is flowing after 0.3 second? After 0.5 second? After 1 second? (e) What is the maximum current? (f) Graph this function 1 = 12(t) on the same coordinate axes as l,(t).

Cl\I'rent in a RC Circuit The equation governing the amount of current 1 (in amperes) after time t (in microseconds) in a single RC circuit consisting of a resistance R (in ohms), a capacitance C (in microfarads), and an electromotive force E (in volts) is 1 = E e-t/(RC) R (a) If E = 120 volts, R = 2000 ohms, and C = 1.0 microfarad, how much current I, is flowing initially (t = O)? After 1 000 microseconds? After 3000 microseconds? (b) What is the maximum current? (c) Graph this function I = 1J (t), measuring 1 along the y-axis and t along the x-axis. (d) If E = 120 volts, R = 1000 ohms, and C = 2.0 microfarads, how much current 1 2 is flowing initially? After 1000 microseconds? After 3000 microseconds? (e) What is the maximum current? (f) Graph this function 1 = 1 2 (t) on the same coordinate axes as 1J (/).

Another Formula for e Use a calculator to compute the values of 1 1 1 2 + -+-+ .. +- 2! 3! n! for n = 4, 6, 8, and 10. Compare each result with e. [Hint: I! = 1, 2! = 2 1, 3! = 3 2 ' 1, n! = n(n - 1) .. (3) (2) (1 ).]

Another Formula for e Use a calculator to compute the various values of the expression. Compare the values to e. 2 + 1 1 + 1 2 + 2 3 + 3 4 + 4 etc.

Difference Quotient If f(x) = aX, show that f (x + h ) - f (x) '--'- 0 a" - 1 '--- --'-----'- = a o' -- h"* 0 h h

If f(x) = a'\ show that f(A + B) = f(A) ' f(B).

H f(x) = a" show that f( -x) = f(x)

If f(x) = a'\ show that f( ax) = [f(x) J"

Problems 115 and 116 provide definitions for two other transcendental functions. Problems 115 and 116 provide definitions for two other transcendental functions. (a) Show that f(x) = sinh x is an odd function. . (b) Graph f(x) = sinh x using a graphing utility .

Problems 115 and 116 provide definitions for two other transcendental functions. The hyperbolic cosine function, designated by cosh x, is defined as 1 ( 0 cosh x = - eX + e-" ) 2 (a) Show that f(x) = cosh x is an even function. (b) Graph f(x) = cosh x using a graphing utility. (c) Refer to Problem 115. Show that, for every x, (cosh x)2 - (sinh X)2 = 1

Historical Problem Pierre de Fermat (1601-1665) conjectured that the function f(x) = 2 (2') + 1 for x = 1, 2, 3, ... , would always have a value equal to a prime number. But Leonhard Euler (1707-1783) showed that this formula fails for x = 5. Use a calculator to determine the prime numbers produced by f for x = 1, 2, 3, 4. Then show that f(5 ) = 641 X 6,700,417, which is not prime.

The bacteria in a 4-liter container double every minute. After 60 minutes the container is full. How long did it take to fill half the container?

Explain in your own words what the number e is. Provide at least two applications that use this number.

Do you think that there is a power function that increases more rapidly than an exponential function whose base is greater than I? Explain.

As the base a of an exponential function f(x) = aX, a > 1, increases, what happens to the behavior of its graph for x > O? What happens to the behavior of its graph for x < O?

The graphs of y = a-x and y = (r are identical. Why?

Solve the inequality: 3 x - 7 :=; 8 - 2x (pp. 128-13 1)

Solve the inequality: x 2 - x - 6 > 0 (pp. 314-316)

Solve the inequality: x -1 > 0 (pp. 369-373) x+4

The domain of the logarithmic function f(x) = log" x is ____.

The graph of every logarithmic function f(x) = log" x, a > 0, a *- 1, passes through three points: __ , __ , and ___.

If the graph of a logarithmic function f(x) = log" x, a > 0, a *- 1, is increasing, then its base must be larger than _____.

True or False If y = log" x, then y = aX.

True or False The graph of f(x) = log" x, a > 0, a *- 1, has an x-intercept equal to 1 and no y-intercept.

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. 9 = 3 2

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. 16 = 4 2

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. a2 = 1 .6

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. a3 = 2.1

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. 2x = 7.2

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. 3x = 4. 6

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. eX = 8

In Problems 9-16, change each exponential expression to an equivalent expression involving a logarithm. e2.2 = M

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. log28 = 3

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. 10g{) = -2

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. log" 3 = 6

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. 10gb 4 = 2

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. log3 2 = x

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. log2 6 = x

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. In 4 = x

In Problems 17-24, change each logarithmic expression to an equivalent expression involving an exponent. In x = 4

In Problems 25-36, find the exact value of each logarithm without using a calculator. log2 1

In Problems 25-36, find the exact value of each logarithm without using a calculator. logs 8

In Problems 25-36, find the exact value of each logarithm without using a calculator. logs 25

In Problems 25-36, find the exact value of each logarithm without using a calculator. 10g 3 (i)

In Problems 25-36, find the exact value of each logarithm without using a calculator. log'!2 16

In Problems 25-36, find the exact value of each logarithm without using a calculator. log'!39

In Problems 25-36, find the exact value of each logarithm without using a calculator. 10g l O V

In Problems 25-36, find the exact value of each logarithm without using a calculator. 10gsV2s

In Problems 25-36, find the exact value of each logarithm without using a calculator. logv'24

In Problems 25-36, find the exact value of each logarithm without using a calculator. 10gy'39

In Problems 25-36, find the exact value of each logarithm without using a calculator. 1nVe

In Problems 25-36, find the exact value of each logarithm without using a calculator. In e 3

In Problems 37-48, find the domain of each function. f(x) = In(x - 3)

In Problems 37-48, find the domain of each function. g(x) = In(x - 1)

In Problems 37-48, find the domain of each function. F(x) = Iog2 x2

In Problems 37-48, find the domain of each function. H(x) = logs x3

In Problems 37-48, find the domain of each function. f(x) = 3 - 210g4[1 - 5]

In Problems 37-48, find the domain of each function. g(x) = 8 + 5 In(2x + 3)

In Problems 37-48, find the domain of each function. f(x) = In(_l_) x + 1

In Problems 37-48, find the domain of each function. g(x) = In( l ) x - 5

In Problems 37-48, find the domain of each function. X + 1 45. g(x) = log5 - ) x

In Problems 37-48, find the domain of each function. hex) = IOg3C 1 )

In Problems 37-48, find the domain of each function. f(x) =

In Problems 37-48, find the domain of each function. g(x) =-l In x

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. In]

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. 3

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. 10 I 0.04

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. In5 3 1 -

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. In 4 + In 2 ---- log4 + log2

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. log 15 + log 20 --=----= In 15 + In 20

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. 21n 5 + log 50 ----=-- log4 - In 2

In Problems 49-56, use a calculator to evaluate each expression. Round your answer to three decimal places. 310g 80 - In 5 -=-----.,-- log 5 + In 20

Find a so that the graph of f( x) = log" x contains the point ( 2, 2).

Find a so that the graph of f(x) = log" x contains the point (, - 4).

In Problems 59-62, graph each function and its inverse on the same Cartesian plane. f(x) = 3'\ r' (x) = log3 x

In Problems 59-62, graph each function and its inverse on the same Cartesian plane. f(x) = 4\rl(x) = log4 x

In Problems 59-62, graph each function and its inverse on the same Cartesian plane. f(x) = "2 ;r1(x) = logl x

In Problems 59-62, graph each function and its inverse on the same Cartesian plane. f(x) = ;rl(x) = logp:

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x 3 x = 0 1 x -

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x 3 :x = 1 I -1 -3

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x YI o

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x Yt 3 x=O ,.. 5 -1 t x -3

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x Y 3 x=O ----'---- -1

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x -1 5x

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x Y 3 rx= 0 -1 -3 -3

In Problems 63-70, the graph of a logarithmic function is given. Match each graph to on e of the following functions: A. Y = log3 X B. Y = IOg3( -x) c. y = -log3x D. Y = -log3( -x) E. Y = IOg3 x-I F. y = IOg3(X - 1) G. Y = log3(1 - x) H. y = 1 - log3 x ytx= 1 3 -3

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = In(x + 4)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = In(x - 3)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = 2 + In x

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = -In( -x)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = In(2x) - 3

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = -21n (x + 1)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = log(x - 4) + 2

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = -Iogx - 5

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = "2 log(2x)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = loge -2x)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = 3 + log3(X + 2)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = 2 - IOg3(X + 1)

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = ex+2 - 3

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = 3ex + 2

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = 2 x/ 3 + 4

In Problems 71-86, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find f1, the inverse of f (e) Use f-1 to find the range of f (f) Graph f-1. f(x) = -3 x +1

In Problems 87-110, solve each equation. log3 x = 2

In Problems 87-110, solve each equation. log5 x = 3

In Problems 87-110, solve each equation. IOg2 (2x + 1) = 3

In Problems 87-110, solve each equation. log3(3x - 2) = 2

In Problems 87-110, solve each equation. logx 4 = 2

In Problems 87-110, solve each equation. IOgx () = 3

In Problems 87-110, solve each equation. In eX = 5

In Problems 87-110, solve each equation. In e-2x = 8

In Problems 87-110, solve each equation. IOg464 = x

In Problems 87-110, solve each equation. logs 625 = x

In Problems 87-110, solve each equation. log3243 = 2x + 1

In Problems 87-110, solve each equation. log636 = 5x + 3

In Problems 87-110, solve each equation. e3x = 10

In Problems 87-110, solve each equation. e - 2x = .!. 3

In Problems 87-110, solve each equation. e2x+S = 8

In Problems 87-110, solve each equation. e -2x+1 = 13

In Problems 87-110, solve each equation. log3(X2 + 1) = 2

In Problems 87-110, solve each equation. logs(x 2 + x + 4) = 2

In Problems 87-110, solve each equation. IOg2 8x = -3

In Problems 87-110, solve each equation. log3 3x = -1

In Problems 87-110, solve each equation. 5eO.2x = 7

In Problems 87-110, solve each equation. 8 102x-7 = 3

In Problems 87-110, solve each equation. 2 _102-x = 5

In Problems 87-110, solve each equation. 4 ex+1 = 5

SupposethatG(x) = IOg3(2x + 1). (a) What is the domain of G? (b) What is G(40)? What point is on the graph of G? (c) If G(x) = 2, what is x? What point is on the graph of G?

Suppose that F(x) = IOg2 (X + 1) - 3. (a) What is the domain of F? (b) What is F(7)? What point is on the graph of f? (c) If F(x) = -1, what is x? What point is on the graph of F?

In Problems 113-116, graph each function. Based on the graph, state the domain and the range and find any intercepts. f(x) = { In (-X) if x < 0 In x if x > 0

In Problems 113-116, graph each function. Based on the graph, state the domain and the range and find any intercepts. f (x) = {ln l (- ( X) ) if x :5-1 - n -x if -1 < x < 0

In Problems 113-116, graph each function. Based on the graph, state the domain and the range and find any intercepts. f (x) = In x if 0 < x < 1 if x 2: 1

In Problems 113-116, graph each function. Based on the graph, state the domain and the range and find any intercepts. f (x) = -I nx if 0 < x < 1 if x 2: 1

Chemistry The pH of a chemical solution is given by the formula where [H+ ] is the concentration of hydrogen ions in moles per liter. Values of pH range from 0 (acidic) to 14 (alkaline). (a) What is the pH of a solution for which [H+] is 0.1? (b) What is the pH of a solution for which [H+] is 0.01? (c) What is the pH of a solution for which [H+] is 0.001? (d) What happens to pH as the hydrogen ion concentration decreases? (e) Determine the hydrogen ion concentration of an orange (pH = 3.5). (f) Determine the hydrogen ion concentration of human blood (pH = 7.4).

Diversity Index Shannon's diversity index is a measure of the diversity of a population. The diversity index is given by the formula H = -(PI log PI + P210g P2 + ... + Pl1log PH) where PI is the proportion of the population that is species 1, P2 is the proportion of the population that is species 2, and so on. (a) According to the U.S. Census B ureau, the distribution of race in the United States in 2000 was as follows: Race American Indian or Native Alaskan Asian Black or African American Hispanic Native Hawaiian or Pacific Islander White Source: u.s. Census Bureau Proportion 0.014 0.041 0.128 0.124 0.003 0.690 Compute the diversity index of the United States in 2000. (b) The largest value of the diversity index is given by H max = logeS), where S is the number of categories of race. Compute H max' H (c) The evenness ratio is given by Ef{ = --, where Hmax 0:5 EN :5 1. If Ef{ = 1, there is complete evenness. Compute the evenness ratio for the United States. (d) Obtain the distribution of race for the United States in 1990 from the Census B ureau. Compute Shannon's diversity index. Is the United States becoming more diverse? Why?

Atmospheric Pressure TIle atmospheric pressure p on a balloon or an aircraft decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height h (in kilometers) above sea level by the formula P = 760e-O . 1 45h (a) Find the height of an aircraft if the atmospheric pressure is 320 millimeters of mercury. (b) Find the height of a mountain if the atmospheric pressure is 667 millimeters of mercury.

Healing of Wounds The normal healing of wounds can be modeled by an exponential function. If Ao represents the original area of the wound and if A equals the area of the wound, then the formula A = Aoe - 0.35 11 describes the area of a wound after n days following an injury when no infection is present to retard the healing. Suppose that a wound initially had an area of 100 square millimeters. (a) If healing is taking place, after how many days will the wound be one-half its original size? (b) How long before the wound is 10% of its original size?

Exponential Probability Between 1 2:00 PM and 1:00 PM, cars arrive at Citibank's drive-thru at the rate of 6 cars per hour (0.1 car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within t minutes of 12:00 PM. F(t) = 1 - e-O.II (a) Determine how many minutes are needed for the probability to reach 50%. (b) Determine how many minutes are needed for the probability to reach 80%. (c) Is it possible for the probability to equal 100% ? Explain.

Exponential Probability Between 5:00 PM and 6:00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour (0.15 car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within t minutes of 5:00 PM. F(t) = 1 - e-O . 151 (a) Determine how many minutes are needed for the probability to reach 50%. (b) Determine how many minutes are needed for the probability to reach 80%.

Drug Medication The formula D = 5e-0.4h can be used to find the number of milligrams D of a certain drug that is in a patient's bloodstream h hours after the drug was administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections?

Spreading of Rumors A model for the number N of people in a college community who have heard a certain rumor is N = P(l - e-0 .J5d) where P is the total population of the community and d is the number of days that have elapsed since the rumor began. In a community of 1000 students, how many days will elapse before 450 students have heard the rumor?

Current in a RL Circuit The equation governing the amount of current 1 (in amperes) after time t (in seconds) in a simple RL circuit consisting of a resistance R (in ohms), an inductance L (in henrys), and an electromotive force E (in volts) is I = . [1 - e-(R/L) I] R If E = 12 volts, R = 10 ohms, and L = 5 henrys, how long does it take to obtain a current of 0.5 ampere? Of 1.0 ampere? Graph the equation.

Learning Curve Psychologists sometimes use the function L(t) = A(l - ekl) to measure the amount L learned at time t. The number A represents the amount to be learned, and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student learned 20 vocabulary words after 5 minutes. (a) Determine the rate of learning k. (b) Approximately how many words will the student have learned after 10 minutes? (c) After 15 minutes? (d) How long does it take for the student to learn 180 words?

Loudness of Sound Problems 127-130 use the following discussion: The loudness L( x), measured in decibels, of a sound of intensity x, x measured in watts per square metel; is defined as L(x) = 10 log -, where 10 = 10- 1 2 watt per square meter is the least intense sound 10 that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. Normal conversation: intensity of x = 10- 7 watt per square meter.

Loudness of Sound Problems 127-130 use the following discussion: The loudness L( x), measured in decibels, of a sound of intensity x, x measured in watts per square metel; is defined as L(x) = 10 log -, where 10 = 10- 1 2 watt per square meter is the least intense sound 10 that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. Amplified rock music: intensity of 10- 1 watt per square meter.

Loudness of Sound Problems 127-130 use the following discussion: The loudness L( x), measured in decibels, of a sound of intensity x, x measured in watts per square metel; is defined as L(x) = 10 log -, where 10 = 10- 1 2 watt per square meter is the least intense sound 10 that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. Heavy city traffic: intensity of x = 10- 3 watt per square meter.

Loudness of Sound Problems 127-130 use the following discussion: The loudness L( x), measured in decibels, of a sound of intensity x, x measured in watts per square metel; is defined as L(x) = 10 log -, where 10 = 10- 1 2 watt per square meter is the least intense sound 10 that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. Diesel truck traveling 40 miles per hour 50 feet away: intensity 10 times that of a passenger car traveling 50 miles per hour 50 feet away whose loudness is 70 decibels.

The Richter Scale Problems 131 and 132 use the following discussion: The Richter scale is one way of converting seismographic readings into numbers the Richter scale that provide an easy reference for measuring the magnitude M of an earthquake. All earthquakes are compared to a zero-level earthquake whose seismographic reading measures 0.001 millimeter at a distance of 100 kilometers from the epicenteJ: An earthquake whose seismographic reading measures x millimeters has magnitude M(x), given by M( x) = 10g( ) o where Xo = 10-3 is the reading of a zero-level earthquake the same distance from its epicenter. In Problems 131 and 132, determine the magnitude of each earthquake. Magnitude of an Earthquake Mexico City in 1985: seismographic reading of 125,892 millimeters 100 kilometers from the center.

The Richter Scale Problems 131 and 132 use the following discussion: The Richter scale is one way of converting seismographic readings into numbers the Richter scale that provide an easy reference for measuring the magnitude M of an earthquake. All earthquakes are compared to a zero-level earthquake whose seismographic reading measures 0.001 millimeter at a distance of 100 kilometers from the epicenteJ: An earthquake whose seismographic reading measures x millimeters has magnitude M(x), given by M( x) = 10g( ) o where Xo = 10-3 is the reading of a zero-level earthquake the same distance from its epicenter. In Problems 131 and 132, determine the magnitude of each earthquake. Magnitude of an Earthquake San Francisco in 1906: seismographic reading of 7943 millimeters 100 kilometers from the center.

Alcohol and Driving The concentration of alcohol in a person's bloodstream is measurable. Suppose that the relative risk R of having an accident while driving a car can be modeled by the equation where x is the percent of concentration of alcohol in the bloodstream and k is a constant. (a) Suppose that a concentration of alcohol in the bloodstream of 0.03 percent results in a relative risk of an accident of 1.4. Find the constant k in the equation. (b) Using this value of k, what is the relative risk if the concentration is 0.17 percent? (c) Using the same value of k, what concentration of alcohol corresponds to a relative risk of 100? (d) If the law asserts that anyone with a relative risk of having an accident of 5 or more should not have driving privileges, at what concentration of alcohol in the bloodstream should a driver be arrested and charged with a DUI? (e) Compare this situation with that of Example 10. If you were a lawmaker, which situation would you support? Give your reasons.

Is there any function of the form y = xG', O < a < 1, that increases more slowly than a logarithmic function whose base is greater than I? Explain.

In the definition of the logarithmic function, the base a is not allowed to equal 1. Why?

Critical Thinking In buying a new car, one consideration might be how well the price of the car holds up over time. Different makes of cars have different depreciation rates. One way to compute a depreciation rate for a car is given here. Suppose that the current prices of a certain Mercedes automobile are as follows: Age in Years New 2 3 4 5 $38,000 $36,600 $32,400 $28,750 $25,400 $21 ,200 Use the formula New = Old(eRt) to find R, the annual depreciation rate, for a specific time t. When might be the best time to trade in the car? Consult the NADA ("blue") book and compare two like models that you are interested in. Which has the better depreciation rate?

The logarithm of a product equals the __ of the logarithms.

If logs M = - - 8 ' then M = o ____.

loga Mr = ___.

True or False In(x + 3) - In(2x) = In () 2x

True or False log2(3x4) = 4 log2(3x)

True or False logz16 =--In 2

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: log33

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: log2 T

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: In e -

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: In eV"

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: 2 log27

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: e1n s

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: logs 2 + logs 4

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: log6 9 + log64

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: log6 18 - log63

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: logs 16 - logs 2

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: og2 6 log6 4

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: og3 8 . logs 9

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: 3 log35 - log 34

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: 5log5 6+log57

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: eloge2 16

In Problems 7-22, use properiies of logarithms to find the exact value of each expression. Do not use a calculator: eloge 29

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. In 6

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. In '

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. In 1.5

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. In 0 .5

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. In 8

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. In 27

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. In '#6

In Problems 23-30, suppose that In 2 = a and In 3 = b. Use properties of logarithms to write each logarithm in terms of a and b. ln

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. logs(25x)

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. log3 '

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. log2 z3

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. log7 (x5)

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In(ex)

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In x

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In( xeX)

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. .lnr e'

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. loga (u2 v3) u > 0, v = 0

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. . IOg2(;2 ) a> O,b > 0

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In(x2) 0 < x < 1

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In( x) x> 0

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. g2 --) x > 3 x - 3

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. gs 2 x > 1 x -I

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. log 2 . (x + 3) x> 0

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. log 2 x> 2 (x - 2

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In 2 x> 2 (x + 4)

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In -'--- 2 ---'-- x - I x>4

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In 3 (x - 4) x>4

In Problems 31-50, write each expression as a sum and/or difference of logarithms. Express powers as factors. In [ 5X2? J 4(x + 1 )- O<x<1

In Problems 51-64, write each expression as a single logarithm. 3 logs u + 4 logs v

In Problems 51-64, write each expression as a single logarithm. 2 log3 U - log3 V

In Problems 51-64, write each expression as a single logarithm. log3 Vx - log3 x3

In Problems 51-64, write each expression as a single logarithm. log2 (1) + log2 1

In Problems 51-64, write each expression as a single logarithm. lo(x2 - 1) - 5 Io(x + 1)

In Problems 51-64, write each expression as a single logarithm. log(x2 + 3x + 2) - 2log(x + 1)

In Problems 51-64, write each expression as a single logarithm. In -- + In " ) - In(x2 - 1)

In Problems 51-64, write each expression as a single logarithm. x 2+7X+6 ) log ? - log r-4 x+2

In Problems 51-64, write each expression as a single logarithm. 8log2 V3x - 2 - log2 (4) + log2 4

In Problems 51-64, write each expression as a single logarithm. 21 log3 vx + log3 (9x2) - log3 9

In Problems 51-64, write each expression as a single logarithm. 2Ioga(5xJ) - z loga(2x + 3)

In Problems 51-64, write each expression as a single logarithm. log(x3 + 1) + ?, log(x2 + 1)

In Problems 51-64, write each expression as a single logarithm. 2 log2(x+1) - log2(x+3) - log2(x-1)

In Problems 51-64, write each expression as a single logarithm. 3 logs(3x + 1) - 2 logs(2 x - 1) - logs x

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. log3 21

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. IOg5 18

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. logt/3 71

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. logt/2 15

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. logy12 7

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. logys 8

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. log" e

In Problems 65-72, use the Change-of-Base Form ula and a calc ulator to eval uate each logarithm. Round your answer to three decimal places. log7/" v2

In Problems 73-78, graph each function using a graphing utility and the Change-of-Base Formula. Y = log4 x

In Problems 73-78, graph each function using a graphing utility and the Change-of-Base Formula. Y = logs x

In Problems 73-78, graph each function using a graphing utility and the Change-of-Base Formula. Y = log2( x + 2)

In Problems 73-78, graph each function using a graphing utility and the Change-of-Base Formula. Y = log4( x - 3)

In Problems 73-78, graph each function using a graphing utility and the Change-of-Base Formula. Y = logx-t( x + 1)

In Problems 73-78, graph each function using a graphing utility and the Change-of-Base Formula. y = logx+2( x - 2)

Hf(x) = In x, g( x) = e\ and h(x) = x 2 , find: (a) (f a g) ( x). What is the domain of fog? (b) (g a f)(x). What is the domain of go f? (c) (f a g)(5) (d) (f a h)(x). What is the domain of f a h? (e) (f a h)(e)

Hf(x) = log2 x,g(x) = 2\ and h(x) = 4x, find: (a) (f a g) (x). What is the domain of fog? (b) (g a f)(x). What is the domain of g of? (c) (f a g)(3) (d) (f a h)(x). What is the domain of f a h? (e) (f a h)(8)

In Problems 81-90, express y as a function of x. The constant C is a positive number. In y = In x + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. In y = In( x + C)

In Problems 81-90, express y as a function of x. The constant C is a positive number. In y = In x + In( x + 1) + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. In y = 21n x - In( x + 1) + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. In y = 3x + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. In y = -2x + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. In (y - 3) = -4x + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. In (y + 4) = 5x + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. 31n y = "2ln( 2x + 1) - "3 ln( x + 4) + In C

In Problems 81-90, express y as a function of x. The constant C is a positive number. 21n y = -"2ln x + "3 ln( x 2 + 1) + InC

Find the value of log2 3 log3 4 log4 5 . logs 6 log6 7 log7 8

Find the value of log2 4 log4 6 log6 8

Find the value of log2 3 log3 4 ... Iog,, (n + 1) IOg,,+1 2

Find the value of log2 2 . log2 4 ... . log2 2"

Show that log,,(x + ) + log,,(x -) = o.

Show that log,,( Vx + ) + log,,( Vx -) = o.

Show that In(l + e 2 x) = 2x + In( 1 + e - 2x ).

Difference Quotient If f(x) = log" x, show that h = log" 1 + , h '1= o.

If f(x) = log" x, show that -f(x) = IOgl/" x.

If f(x) = log" x, show that f(AB) = f(A) + feB).

lf f( x) = IOga x, show that f ( ) = -f(x).

If f(x) = log" x, show that f(x") = a f( x).

Show that IOg,,( ) = log" M - log" N, where a, M, and N are positive real numbers and a '1= 1.

Show that IOg,, ( ) = - log" N, where a and N are positive real numbers and a '1= 1.

Graph Yt = log( x 2 ) and Y2 = 2 Iog( x) using a graphing utility. Are they equivalent? What might account for any differences in the two functions?

Write an example that illustrates why ( log" x)' '1= r log" x.

Write an example that illustrates why log2( X + y) '1= log2 X + log2 y.

Does 310g3(-5 ) = -5? Why or why not?

Solve x 2 - 7x - 30 = O. (pp. 97-106)

Solve (x + 3 ) 2 - 4( x + 3 ) + 3 = O. (pp. 11 9-121)

Approximate the solution(s) to x3 = x 2 - 5 using a graphing utility. (pp. AS-AIO)

Approximate the solution(s) to x3 - 2x + 2 = 0 using a graphing utility. (pp. AS-AlO)

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log4x = 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log (x + 6) = 1

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log2 (5 x) = 4

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. Og3 (3 x - 1) = 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. IOg4(X + 2) = IOg4 S

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. IOg5(2x + 3) = log5 3

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. 100, x = 2 100, 2 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. -2 IOg4 x = IOg4 9

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. 31og2 X = - log2 27

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. 2 IOg5 x = 3 IOg5 4

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. 3 IOg2 (X - 1) + IOg2 4 = 5

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. 2 log3(x + 4) - IOg3 9 = 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log x + log(x + 15) = 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log x + log (x - 21 ) = 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log(2x + 1) = 1 + log(x - 2)

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log(2x) - log(x - 3) = 1

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. og2 (x + 7) + Iog2 (X + S) = 1

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log6(X + 4) + log6(X + 3) = 1

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. logs(x + 6) = 1 - logs( x + 4)

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log5(x + 3 ) = 1 - IOg5(X - 1)

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. In x + In(x + 2) = 4

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. In (x + 1) - In x = 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log3(X + 1) + IOg3(X + 4) = 2

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. IOg2(X + 1) + IOg2(X + 7) = 3

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. IOgl/3(X2 + x) - IOg1 /3(X2 - x) = -1

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. IOg4(X2 - 9) - lo(x + 3 ) = 3

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. log{[(x - 1) - log{[(x + 6) = log{[(x - 2) - loga(x + 3)

In Problems 5-32, solve each logarithmic equation. Express irrarional solutions in exact form and as a decimal rounded to 3 decimal places. logll x + 10gaC x - 2) = loga(x + 4)

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2x-5 = S

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 5-x = 25

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2x = 10

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 3x = 14

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. S-x = 1.2

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. TX = 1 .5

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 5(23x) = 8

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 0.3 ( 402X) = 0.2

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 3 1 - 2 x = 4x

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2x+1 = 51- 2 x

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. (_.))"'o)X = 7 ' -x

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. G ),-x = 5x

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 1 .2x = (0.5 fx

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 0.31 + x = 1. 72 x1

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 7T1 -x = eX

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. ex+ 3 = 7Tx

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2 2 x + 2x - 12 = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 32x + 3" - 2 = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 3 2 x + 3x+ I - 4 = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2 2x + 2x+ 2 - 12 = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 1 6x + 4x+1 - 3 = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 9x - 3x+ 1 + 1 = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 25x - S 5x = -16

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 36x - 6 6x = -9

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 3 4x + 4 2x + S = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2 49x + 1 1 7x + 5 = 0

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 4x - 10 4-x = 3

In Problems 33-60, solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 3"' - 14 Tx = 5

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. logs(x + 1) - IOg4(X - 2) = 1

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. IOg2 (X - 1) - IOg6(X + 2) = 2

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. eX = -x

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. e 2 x = x + 2

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. ex = x2

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. ex = x3

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. In x = -x

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. In(2x) = -x + 2

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. In x = x3 - 1

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. ln x = -x2

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. eX + In x = 4

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. eX - In x = 4

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. e-x = In x

In Problems 61-74, use a graphing utility to solve each equation. Express your answer rounded to two decimal places. e-x = -In x

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log2(x + 1) - log4 X = 1 [Hint: Change log4 x to base 2.]

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log2(3x + 2) - log4 X = 3

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. IOg16 x + log4 X + log2 X = 7

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log9 x + 3 log3 X = 14

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. ()2-X = 2X 2

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log2 xlog2 x = 4

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. eX + e-x 81. = 1 2 [Hint: Multiply each side by eX .]

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. ex + e-x = 3

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. ex - e-x = 2

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. eX - e-x = -2

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log5 x + log3 X = 1 [Hint: Use the Change-of-Base Formula and factor out In x. ]

In Problems 75-86, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log2 x + log6 x = 3

f(x) = log2 (x + 3) and g(x) = log2(3x + 1 ) . (a) Solve f(x) = 3. What point is on the graph of f? (b) Solve g(x) = 4. What point is on the graph of g? (c) Solve f(x) = g(x). Do the graphs off and g intersect? If so, where? (d) Solve f(x) + g(x) = 7. (e) Solve f(x) - g(x) = 2.

[( x) = log3(x + 5) and g(x) = log3(x - 1 ). (a) Solve f(x) = 2. What point is on the graph of f? (b) Solve g(x) = 3. What point is on the graph of g? (c) Solve f(x) = g(x). Do the graphs of f and g intersect? If so, where? (d) Solve f(x) + g(x) = 3. (e) Solve f(x) - g(x) = 2.

(a) If f(x) = y+1 and g(x) = 2x+ 2 , graph f and g on the same Cartesian plane. (b) Find the point(s) of intersection of the graphs offand g by solvingf(x) = g(x) . Round answers to three decimal places. Label this point on the graph drawn in part (a). (c) Based on the graph, solve f(x) > g(x).

(a) If f(x) = 5x-1 and g(x) = 2x+ l , graph f and g on the same Cartesian plane. (b) Find the point(s) of intersection of the graphs of f and g by solving f(x) = g(x). Label this point on the graph drawn in part (a). (c) Based on the graph, solve f(x) > g(x).

( a) Graph f (x) = 3x and g( x) = 10 on the same Cartesian plane. (b) Shade the region bounded by the y-axis, f(x) = 3" and g(x) = 10 on the graph drawn in part (a). (c) Solve f(x) = g(x) and label the point of intersection on the graph drawn in part (b).

(a) Graph f(x) = 2x and g(x) = 12 on the same Cartesian plane. (b) Shade the region bounded by the y-axis,/ (x) = 2-', and g(x) = 12 on the graph drawn in part (a). (c) Solve f(x) = g(x) and label the point of intersection on the graph drawn in part (b).

(a) Graph f(x) = 2x+1 and g(x) = Tx+2 on the same Cartesian plane. (b) Shade the region bounded by the y-axis, f(x) = 2x+\ and g(x) = Tx+2 on the graph draw in part (a). (c) Solve f(x) = g(x) and label the point of intersection on the graph drawn in part (b).

(a) Graph f(x) = rx+1 and g(x) = y2 on the same Cartesian plane. (b) Shade the region bounded by the y-axis,f (x) = rr+ l , and g(x) = 3x2 on the graph draw in part (a). (c) Solve f(x) = g(x) and label the point of intersection on the graph drawn in part (b).

(a) Graph I(x) = 2x - 4. (b) Based on the graph, solve f(x) < O.

(a) Graph g(x) = 3x - 9. (b) Based on the graph, solve g( x) > O.

A Population Model The resident popUlation of the United States in 2006 was 298 million people and was growing at a rate of 0.9% per year. Assuming that this growth rate continues, the model pet) = 298(1.009)'- 2 00 6 represents the population P (in millions of people) in year t. (a) According to this model, when will the population of the United States be 310 million people? (b) According to this model, when will the population of the United States be 360 million people? Source: Statistical Abstract of t he United States, 1 25th ed., 2006.

A Population Model The population of the world in 2006 was 6.53 billion people and was growing at a rate of 1.14% per year. Assuming that this growth rate continues, the model pet) = 6.53(1.0114y-2oo 6 represents the population P (in billions of people) in year t. (a) According to this model, when will the population of the world be 9.25 billion people? (b) According to this model, when will the population of the world be 11.75 billion people? Source: u.s. Census Bu reau.

Depreciation The value V of a Chevy Cobalt that is t years old can be modeled by Vet) = 14,512(0.82t (a) According to the model, when will the car be worth $9000? (b) According to the model, when will the car be worth $4000? (c) According to the model , when will the car be worth $2000? Source: Kelley Blue Book

Depreciation The value V of a Dodge Stratus that is t years old can be modeled by Vet) = 19,282(0.84t (a) According to the model, when will the car be worth $15,000? (b) According to the model, when will the car be worth $8000? (c) According to the model, when will the car be worth $2000? Source: Kelley Blue Book

Fill in reasons for each step in the following two solutions. Solve: log3(x - 1 )2 = 2 Solution A log3(x - 1 )2 = 2 (x - 1 )2 = 32 = 9 __ (x - 1) = 3 __ x-I = -3 orx - 1 = 3 x = -2 orx = 4 Solution B log3(x - 1 )2 = 2 2 10g3(x - 1) = 2 __ log3(x - 1) = 1 __ x - l = 3 i = 3 x=4 (a) According to the model, when will the car be worth $15,000? (b) According to the model, when will the car be worth $8000? (c) According to the model, when will the car be worth $2000? Source: Kelley Blue Book Both solutions given in Solution A check. Explain what caused the solution x = -2 to be lost in Solution B.

What is the interest due if $500 is borrowed for 6 months at a simple interest rate of 6% per annum? (pp. 141-142)

If you borrow $5000 and, after 9 months, pay off the loan in the amount of $5500, what per annum rate of interest was charged? (pp. 141-142)

In Problems 3-12, find the amount that results from each investment. $100 invested at 4% compounded quarterly after a period of 2 years

In Problems 3-12, find the amount that results from each investment. $50 invested at 6% compounded monthly after a period of 3 years

In Problems 3-12, find the amount that results from each investment. $500 invested at 8% compounded quarterly after a period of 1 2'2 years

In Problems 3-12, find the amount that results from each investment. $300 invested at 12% compounded monthly after a period of 1 1 '2 years

In Problems 3-12, find the amount that results from each investment. $600 invested at 5% compounded daily after a period of 3 years

In Problems 3-12, find the amount that results from each investment. $700 invested at 6% compounded daily after a period of 2 years

In Problems 3-12, find the amount that results from each investment. $10 invested at 11 % compounded continuously after a period of 2 years

In Problems 3-12, find the amount that results from each investment. $40 invested at 7% compounded continuously after a period of 3 years

In Problems 3-12, find the amount that results from each investment. $100 invested at 10% compounded continuously after a period of 2 years

In Problems 3-12, find the amount that results from each investment. $100 invested at 12% compounded continuously after a period of 3 % years

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $100 after 2 years at 6% compounded monthly

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $75 after 3 years at 8% compounded quarterly

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $1000 after 2 years at 6% compounded daily

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $800 after 3'2 years at 7% compounded monthly

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $600 after 2 years at 4% compounded quarterly

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $300 after 4 years at 3% compounded daily

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $80 after 3 years at 9% compounded continuously

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $800 after 2 years at 8% compounded continuously

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $400 after 1 year at 10% compounded continuously

In Problems 13-22, find the principal needed now to get each amount; that is, find the present value. To get $ 1000 after 1 year at 12% compounded continuously

In Problems 23-26, which of the two rates would yield the larger amount in 1 year? [Hint: Start with a principal of $10,000 in each instance. ] 6% compounded quarterly or 6"4 % compounded annually

In Problems 23-26, which of the two rates would yield the larger amount in 1 year? [Hint: Start with a principal of $10,000 in each instance. ] 9% compounded quarterly or 9"4 % compounded annually

In Problems 23-26, which of the two rates would yield the larger amount in 1 year? [Hint: Start with a principal of $10,000 in each instance. ] 9% compounded monthly or 8.8% compounded daily

In Problems 23-26, which of the two rates would yield the larger amount in 1 year? [Hint: Start with a principal of $10,000 in each instance. ] 8% compounded semiannually or 7.9% compounded daily

In Problems 27-30, find the effective rate of interest. For 5% compounded quarterly

In Problems 27-30, find the effective rate of interest. For 6% compounded monthly

In Problems 27-30, find the effective rate of interest. For 5% compounded continuously

In Problems 27-30, find the effective rate of interest. For 6% compounded continuously

What rate of interest compounded annually is required to double an investment in 3 years?

What rate of interest compounded annually is required to double an investment in 6 years?

What rate of interest compounded annually is required to triple an investment in 5 years?

What rate of interest compounded annually is required to triple an investment in 10 years?

(a) How long does it take for an investment to double in value if it is invested at 8% compounded monthly? (b) How long does it take if the interest is compounded continuously?

(a) How long does it take for an investment to triple in value if it is invested at 6% compounded monthly? (b) How long does it take if the interest is compounded continuously?

What rate of interest compounded quarterly will yield an effective interest rate of 7%?

What rate of interest compounded continuously will yield an effective interest rate of 6%?

Time Required to Reach a Goal IfTanisha has $100 to invest at 8% per annum compounded monthly, how long will it be before she has $150? If the compounding is continuous, how long will it be?

Time Required to Reach a Goal If Angela has $100 to invest at 10% per annum compounded monthly, how long will it be before she has $175? If the compounding is continuous, how long will it be?

Time Required to Reach a Goal How many years will it take for an initial investment of $ 10,000 to grow to $25,000? Assume a rate of interest of 6% compounded continuously.

Time Required to Reach a Goal How many years will it take for an initial investment of $25,000 to grow to $80,000? Assume a rate of interest of 7% compounded continuously.

Price Appreciation of Homes What will a $90,000 house cost 5 years from now if the price appreciation for homes over that period averages 3% compounded annually?

Credit Card Interest Sears charges 1.25% per month on the unpaid balance for customers with charge accounts (interest is compounded monthly). A customer charges $200 and does not pay her bill for 6 months. What is the bill at that time?

Saving for a Car Jerome will be buying a used car for $15,000 in 3 years. How much money should he ask his parents for now so that, if he invests it at 5% compounded continuously, he will have enough to buy the car?

Paying off a Loan John will require $3000 in 6 months to pay off a loan that has no prepayment privileges. If he has the $3000 now, how much of it should he save in an account paying 3% compounded monthly so that in 6 months he will have exactly $3000?

Return on a Stock George is contemplating the purchase of 100 shares of a stock selling for $15 per share. The stock pays no dividends. The history of the stock indicates that it should grow at an annual rate of 15% per year. How much will the 100 shares of stock be worth in 5 years?

Return on an Investment A business purchased for $650,000 in 2001 is sold in 2004 for $850,000. What is the annual rate of return for this investment?

Comparing Savings P lans Jim places $1000 in a bank account that pays 5.6% compounded continuously. After 1 year, will he have enough money to buy a computer system that costs $1060? If another bank will pay Jim 5.9% compounded monthly, is this a better deal?

Savings Plans On January 1, Kim places $1000 in a certificate of deposit that pays 6.8% compounded continuously and matures in 3 months. Then Kim places the $1000 and the interest in a passbook account that pays 5.25% compounded monthly. How much does Kim have in the passbook account on May I?

Comparing IRA Investments Will invests $2000 in his IRA in a bond trust that pays 9% interest compounded semiannually. His friend Henry invests $2000 in his IRA in a certificate 1 of deposit that pays 8"2 % compounded continuously. Who has more money after 20 years, Will or Henry?

Comparing Two Alternatives Suppose that April has access to an investment that will pay 10% interest compounded continuously. Which is better: to be given $1000 now so that she can take advantage of this investment opportunity or to be given $1325 after 3 years?

College Costs The average cost of college at 4-year private colleges was $29,026 in 2005. This was a 5.5% increase from the previous year. Source: Trends in College Pricing 2005, College Board (a) If the cost of college increases by 5.5% each year, what will be the average cost of college at a 4-year private college in 2015? (b) College savings plans, such as a 529 plan, allow individuals to put money aside now to help pay for college later. If one such plan offers a rate of 4% compounded continuously, how much should be put in a college savings plan in 2005 to pay for 1 year of the cost of college at a 4-year private college in 2015?

Analyzing Interest Rates on a Mortgage Colleen and Bill have just purchased a house for $650,000, with the seller holding a second mortgage of $1 00,000. They promise to pay the seller $100,000 plus all accrued interest 5 years from now. The seller offers them three interest options on the second mortgage: (a) Simple interest at 12% per annum (b) 11 % interest compounded monthly (c) 11 % interest compounded continuously Which option is best; that is, which results in the least interest on the loan?

Federal Deficit At the end of fiscal year 2005, the federal budget deficit was $319 billion. At that time, 20-year Series EE bonds had a fixed rate of 3.2% compounded semiannually. If the federal government financed this deficit through EE bonds, how much would it have to pay back in 2025? Source: Us. Treasury Department

Federal Deficit On February 6, 2006, President Bush proposed the fiscal year 2007 federal budget. The proposal projected a fiscal year 2006 deficit of $423 billion and a fiscal year 2007 deficit of $354 billion. Assuming the deficit decreases at the same rate each year, when will the deficit be cut to $100 billion? Source: Office of Management and Budget

Inflation Problems 57-62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. For example, suppose the annual inflation rate is 3%. Then $1000 worth of purchasing power now will have only $970 worth of purchasing power in one year because 3% of the original $1000 (0.03 X 1000 = 30) has been eroded due to inflation. In general, if the rate of inflation averages r% over n years, the amount A that $P will purchase after n years is A = p. (1 - r)" where r is expressed as a decimal. Intlation If the inflation rate averages 3%, how much will $1 000 purchase in 2 years?

Inflation Problems 57-62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. For example, suppose the annual inflation rate is 3%. Then $1000 worth of purchasing power now will have only $970 worth of purchasing power in one year because 3% of the original $1000 (0.03 X 1000 = 30) has been eroded due to inflation. In general, if the rate of inflation averages r% over n years, the amount A that $P will purchase after n years is A = p. (1 - r)" where r is expressed as a decimal. Inflation If the inflation rate averages 2%, how much will $ 1000 purchase in 3 years?

Inflation Problems 57-62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. For example, suppose the annual inflation rate is 3%. Then $1000 worth of purchasing power now will have only $970 worth of purchasing power in one year because 3% of the original $1000 (0.03 X 1000 = 30) has been eroded due to inflation. In general, if the rate of inflation averages r% over n years, the amount A that $P will purchase after n years is A = p. (1 - r)" where r is expressed as a decimal. lnDation If the amount that $1000 will purchase is only $950 after 2 years, what was the average inflation rate?

Inflation Problems 57-62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. For example, suppose the annual inflation rate is 3%. Then $1000 worth of purchasing power now will have only $970 worth of purchasing power in one year because 3% of the original $1000 (0.03 X 1000 = 30) has been eroded due to inflation. In general, if the rate of inflation averages r% over n years, the amount A that $P will purchase after n years is A = p. (1 - r)" where r is expressed as a decimal. Inflation If the amount that $1000 will purchase is only $930 after 2 years, what was the average inflation rate?

Inflation Problems 57-62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. For example, suppose the annual inflation rate is 3%. Then $1000 worth of purchasing power now will have only $970 worth of purchasing power in one year because 3% of the original $1000 (0.03 X 1000 = 30) has been eroded due to inflation. In general, if the rate of inflation averages r% over n years, the amount A that $P will purchase after n years is A = p. (1 - r)" where r is expressed as a decimal. InDation If the average inflation rate is 2 %, how long is it until purchasing power is cut in half?

Inflation Problems 57-62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. For example, suppose the annual inflation rate is 3%. Then $1000 worth of purchasing power now will have only $970 worth of purchasing power in one year because 3% of the original $1000 (0.03 X 1000 = 30) has been eroded due to inflation. In general, if the rate of inflation averages r% over n years, the amount A that $P will purchase after n years is A = p. (1 - r)" where r is expressed as a decimal. Inflation If the average inflation rate is 4%, how long is it until purchasing power is cut in half?

Problems 63-66 involve zero-coupon bonds. A zero-coupon bond is a bond that is sold now at a discount and will pay its face value at the lime when it matures; no interest payments are made. Zero-Coupon Bonds A zero-coupon bond can be redeemed in 20 years for $ 10,000. How much should you be willing to pay for it now if you want a return of: (a) 10% compounded monthly? (b) 10% compounded continuously?

Problems 63-66 involve zero-coupon bonds. A zero-coupon bond is a bond that is sold now at a discount and will pay its face value at the lime when it matures; no interest payments are made. Zero-Coupon Bonds A child's grandparents are considering buying a $40,000 face value zero-coupon bond at birth so that she will have enough money for her college education 17 years later. If they want a rate of return of 8% compounded annually, what should they pay for the bond?

Problems 63-66 involve zero-coupon bonds. A zero-coupon bond is a bond that is sold now at a discount and will pay its face value at the lime when it matures; no interest payments are made. Zero-Coupon Bonds How much should a $ 10,000 face value zero-coupon bond, maturing in 10 years, be sold for now if its rate of return is to be 8% compounded annually?

Problems 63-66 involve zero-coupon bonds. A zero-coupon bond is a bond that is sold now at a discount and will pay its face value at the lime when it matures; no interest payments are made. Zero-Coupon Bonds If Pat pays $12,485.52 for a $25,000 face value zero-coupon bond that matures in 8 years, what is his annual rate of return?

Time to Double or Triple an Investment The formula In m t =----- n In(1 + ) n can be used to find the number of years t required to multiply an investment m times when r is the per annum interest rate compounded n times a year. (a) How many years will it take to double the value of an IRA that compounds annually at the rate of 12%? (b) How many years will it take to triple the value of a savings account that compounds quarterly at an annual rate of 6%? (c) Give a derivation of this formula.

Time to Reach an Investment Goal The formula In A - In P t = ----- r can be used to find the number of years t required for an investment P to grow to a value A when compounded continuously at an annual rate r. (a) How long will it take to increase an initial investment of $1 000 to $8000 at an annual rate of 1O%? (b) What annual rate is required to increase the value of a $2000 IRA to $30,000 in 35 years? (c) Give a derivation of this formula.

Problems 69-72, require the following discussion. The Consumer Price Index (CPI) indicates the relative change in price over time for a fixed basket of goods and services. It is a cost of living index that helps measure the effect of inflation on the cost of goods and services. The CPI Llses the base period 1982-1984 for comparison (the CPI for this period is ZOO). The CPI for January 2006 was ]98.3. This means that $100 in the period 1982-1984 had the same purchasing power as $198.30 in January 2006. In general, if the rate of inflation averages r% over n years, then the CPI index after n years is CPI = CPIo( 1 + 10 )" where CPlo is the CPI index at the beginning of the n-year period. Source: u.s. Bureau of Labor Statistics Consumer Price Index (a) The cpr was 1 52.4 for 1 995 and 195.3 for 2005. Assuming that annual inflation remained constant for this time period, determine the average annual inflation rate. (b) Using the inflation rate from part (a), in what year will the cpr reach 300?

Problems 69-72, require the following discussion. The Consumer Price Index (CPI) indicates the relative change in price over time for a fixed basket of goods and services. It is a cost of living index that helps measure the effect of inflation on the cost of goods and services. The CPI Llses the base period 1982-1984 for comparison (the CPI for this period is ZOO). The CPI for January 2006 was ]98.3. This means that $100 in the period 1982-1984 had the same purchasing power as $198.30 in January 2006. In general, if the rate of inflation averages r% over n years, then the CPI index after n years is CPI = CPIo( 1 + 10 )" where CPlo is the CPI index at the beginning of the n-year period. Source: u.s. Bureau of Labor Statistics Consumer Price Index If the current cpr is 234.2 and the average annual inflation rate is 2.8%, what will be the cpr in 5 years?

Problems 69-72, require the following discussion. The Consumer Price Index (CPI) indicates the relative change in price over time for a fixed basket of goods and services. It is a cost of living index that helps measure the effect of inflation on the cost of goods and services. The CPI Llses the base period 1982-1984 for comparison (the CPI for this period is ZOO). The CPI for January 2006 was ]98.3. This means that $100 in the period 1982-1984 had the same purchasing power as $198.30 in January 2006. In general, if the rate of inflation averages r% over n years, then the CPI index after n years is CPI = CPIo( 1 + 10 )" where CPlo is the CPI index at the beginning of the n-year period. Source: u.s. Bureau of Labor Statistics Consumer Price Index If the average annual inflation rate is 3.1 %, how long will it take for the cpr index to double?

Problems 69-72, require the following discussion. The Consumer Price Index (CPI) indicates the relative change in price over time for a fixed basket of goods and services. It is a cost of living index that helps measure the effect of inflation on the cost of goods and services. The CPI Llses the base period 1982-1984 for comparison (the CPI for this period is ZOO). The CPI for January 2006 was ]98.3. This means that $100 in the period 1982-1984 had the same purchasing power as $198.30 in January 2006. In general, if the rate of inflation averages r% over n years, then the CPI index after n years is CPI = CPIo( 1 + 10 )" where CPlo is the CPI index at the beginning of the n-year period. Source: u.s. Bureau of Labor Statistics Consumer Price Index The base period for the CPI changed in 1998. Under the previous weight and item structure, the cpr for 1 995 was 456.5. If the average annual inflation rate was 5.57%, what year was used as the base period for the CPI?

Explain in your own words what the term compound interest means. What does continuous compounding mean?

Explain in your own words the meaning of present value.

Critical Thinking You have just contracted to buy a house and will seek financing in the amount of $1 00,000. You go to several banks. Bank 1 will lend you $ 1 00,000 at the rate of 8.75% amortized over 30 years with a loan origination fee of 1.75%. Bank 2 will lend you $ 100,000 at the rate of 8.375% amortized over 15 years with a loan origination fee of 1 .5%. Bank 3 will lend you $100,000 at the rate of 9.125% amortized over 30 years with no loan origination fee. Bank 4 will lend you $100,000 at the rate of 8.625% amortized over 15 years with no loan origination fee. Which loan would you take? Why? Be sure to have sound reasons for your choice. Use the information in the table to assist you. If the amollnt of the monthly payment does not matter to you, which loan would you take? Again, have sound reasons for your choice. Compare your final decision with others in the class. Discllss. Monthly Loan Payment Origination Fee Bank 1 $786.70 $1,750.00 Bank 2 $977.42 $1,500.00 Bank 3 $813.63 $0.00 Bank 4 $990.68 $0.00

Growth of an Insect Population The size P of a certain insect population at time t (in days) obeys the function P(t) = 500eoo21. (a) Determine the number of insects at t = 0 days. (b) What is the growth rate of the insect population? (c) What is the population after 10 days? (d) When will the insect population reach 800? (e) When will the insect population double?

Growth of Bacteria The number N of bacteria present in a culture at time t (in hours) obeys the law of uninhibited growth N(t) = 1 000eo.oll (a) Determine the number of bacteria at t = 0 hours. (b) What is the growth rate of the bacteria? (c) What is the population after 4 hours? (d) When will the number of bacteria reach 1 700? (e) When will the number of bacteria double?

Radioactive Decay Strontium 90 is a radioactive material that decays according to the function A(t) = Ao e-o.o2441, where A o is the initial amount present and A is the amount present at time t (in years). Assume that a scientist has a sample of 500 grams of strontium 90. (a) What is the decay rate of strontium 90? (b) How much strontium 90 is left after 10 years? (c) When will 400 grams of strontium 90 be left? (d) What is the half-life of strontiulll 90?

Radioactive Decay Iodine 131 is a radioactive material that decays according to the function A(I) = Ao e-o.oS?I, where A o is the initial amount present and A is the amount present at time 1 (in days). Assume that a scientist has a sample of 100 grams of iodine 1 3 l. (a) What is the decay rate of iodine 131? (b) How much iodine 131 is left after 9 days? (c) When will 70 grams of iodine 131 be left? (d) What is the half-life of iodine 131?

Growth of a Colony of Mosquitoes The population of a colony of Illosquitoes obeys the law of uninhibited growth. (a) If N is the population of the colony and t is the time in days, express N as a function of I. (b) If there are 1 000 mosquitoes initially and there are 1 800 after 1 day, what is the size of the colony after 3 days? (c) How long is it until there are 1 0,000 mosquitoes?

Bacterial Growth A culture of bacteria obeys the law of uninhibited growth. (a) If N is the number of bacteria in the culture and 1 is the time in hours, express N as a function of t. (b) If 500 bacteria are present initially and there are 800 after 1 hour, how many will be present in the culture after 5 hours? (c) How long is it until there are 20,000 bacteria?

Population Growth The population of a southern city follows the exponential law. (a) If N is the population of the city and t is the time in years, express N as a function of t. (b) If the population doubled in size over an 18-month period and the current population is 10,000, what will the population be 2 years from now?

Population Decline The population of a midwestern city follows the exponential law. (a) If N is the population of the city and t is the time in years, express N as a function of t. (b) If the population decreased from 900,000 to 800,000 from 2003 to 2005, what will the population be in 2007?

Radioactive Decay The half-life of radium is 1690 years. If 10 grams are present now, how much will be present in 50 years?

Radioactive Decay The half-life of radioactive potassium is 1 .3 billion years. If 10 grams are present now, how much will be present in 100 years? In 1 000 years?

Estimating the Age of a Tree A piece of charcoal is found to contain 30% of the carbon 14 that it originally had. When did the tree die from which the charcoal came? Use 5600 years as the half-life of carbon 14.

Estimating the Age of a Fossil A fossilized leaf contains 70% of its normal amount of carbon 1 4. How old is the fossil?

Cooling Time of a Pizza Pan A pizza pan is removed at 5:00 PM from an oven whose temperature is fixed at 450F into a room that is a constant 70F. After 5 minutes, the pan is at 300F. (a) At what time is the temperature of the pan 135F? (b) Determine the time that needs to elapse before the pan is 160F. (c) What do you notice about the temperature as time passes?

Newton's Law of Cooling A thermometer reading nOF is placed in a refrigerator where the temperature is a constant 38F. (a) If the thermometer reads 60F after 2 minutes, what will it read after 7 minutes? (b) How long will it take before the thermometer reads 39F? (c) Determine the time needed to elapse before the thermometer reads 45F. (d) What do you notice about the temperature as time passes?

Newton's Law of Heating A thermometer reading 8C is brought into a room with a constant temperature of 35C. If the thermometer reads 15C after 3 minutes, what will it read after being in the room for 5 minutes? For 10 minutes? [Hint: You need to construct a formula similar to equation (4).]

Warming Time of a Beerstein A beerstein has a temperature of 28F. It is placed in a room with a constant temperature of 70F. After 10 minutes, the temperature of the stein has risen to 35F. What will the temperature of the stein be after 30 minutes? How long will it take the stein to reach a temperature of 45F? (See the hint given for Problem 15.)

Decomposition of Chlorine in a Pool Under certain water conditions, the free chlorine (hypochlorous acid, HOCl) in a swimming pool decomposes according to the law of uninhibited decay. After shocking his pool, Ben tested the water and found the amount of free chlorine to be 2.5 parts per million (ppm). Twenty-four hours later, Ben tested the water again and found the amount of free chlorine to be 2.2 ppm. What will be the reading after 3 days (that is, n hours)? When the chlorine level reaches 1 .0 ppm, Ben must shock the pool again. How long can Ben go before he must shock the pool again?

Decomposition of Dinitrogen Pentoxide At 45C, dinitrogen pentoxide (N205) decomposes into nitrous dioxide (N02) and oxygen (02) according to the law of uninhibited decay. An initial amount of 0.25 M of dinitrogen pentoxide decomposes to 0.15 M in 17 minutes. How much dinitrogen pentoxide will remain after 30 minutes? How long will it take until 0.01 M of dinitrogen pentoxide remains?

Decomposition of Sucrose Reacting with water in an acidic solution at 35C, sucrose (C12H2201 1) decomposes into glucose (C6H1 z06) and fructose (C6H1 206)* according to the law of uninhibited decay. An initial amount of 0.40 M of sucrose decomposes to 0.36 M in 30 minutes. How much sucrose will remain after 2 hours? How long will it take until 0.10 M of sucrose remains?

Decomposition of Salt in Water Salt (NaCl) decomposes in water into sodium (Na+) and chloride (Cl-) ions according to the law of uninhibited decay. If the initial amount of salt is 25 kilograms and, after 10 hours, 15 kilograms of salt is left, how much salt is left after 1 day? How long does it take until kilogram of salt is left?

Radioactivit)1 from Chernobyl After the release of radioactive material into the atmosphere from a nuclear power plant at Chernobyl (Ukraine) in 1986, the hay in Austria was contaminated by iodine 131 (half-life 8 days). If it is safe to feed the hay to cows when 10% of the iodine 131 remains, how long did the farmers need to wait to use this hay?

Pig Roasts The hotel Bora-Bora is having a pig roast. At noon, the chef put the pig in a large earthen oven. The pig's original temperature was 75F. At 2:00 PM the chef checked the pig's temperature and was upset because it had reached ':' Author's Note: Surprisingly, the chemical formulas for glucose and fructose are the same. This is not a typo. only lOOF. If the oven's temperature remains a constant 325F, at what time may the hotel serve its guests, assuming that pork is done when it reaches 1 75F?

Proportion of the Population That Owns a DVD Player The logistic growth model P( 0.9 t) = 1 + 6e -O.321 relates the proportion of U.S. households that own a DVD player to the year. Let I = 0 represent 2000, I = 1 represent 2001, and so on. (a) Determine the maximum proportion of households that will own a DVD player. (b) What proportion of households owned a DVD player in 2000 (t = OJ? (c) What proportion of households owned a DVD player in 2005 (t = 5)? (d) When will 0.8 (80%) of U.S. households own a DVD player?

Market Penetration of Intel's Coprocessor The logistic growth model 0.90 pet) = 1 + 3.5e-O.3391 relates the proportion of new personal computers (pes) sold at Best Buy that have Intel's latest coprocessor t months after it has been introduced. (a) Determine the maximum proportion of pes sold at Best Buy that will have Intel's latest coprocessor. (b) What proportion of computers sold at Best Buy will have Intel's latest coprocessor when it is first introduced (t = OJ? (c) What proportion of pes sold will have Intel's latest coprocessor t = 4 months after it is introduced? (d) When will 0.75 (75%) of pes sold at Best Buy have Intel's latest coprocessor? (e) How long will it be before 0.45 (45%) of the pes sold by Best Buy have Intel's latest coprocessor?

Population of a Bacteria Culture The logistic growth model 1000 pet) = 1-+ 3-2-.33-e--:: . 0437"91 represents the population (in grams) of a bacterium after I hours. (a) Determine the carrying capacity of the environment. (b) What is the growth rate of the bacteria? (c) Determine the initial population size. (d) What is the population after 9 hours? (e) When will the population be 700 grams? (E) How long does it take for the population to reach onehalf the carrying capacity?

Population of an Endangered Species Often environmentalists capture an endangered species and transport the species to a controlled environment where the species can produce offspring and regenerate its population. Suppose that six American bald eagles are captured, transported to Montana, and set free. Based on experience, the environmentalists expect the population to grow according to the model P(I) _ 500 - 1 + 83.33eO.1621 where t is measured in years. (a) Determine the carrying capacity of the environment. (b) What is the growth rate of the bald eagle? (c) What is the population after 3 years? (d) When will the population be 300 eagles? (e) How long does it take for the population to reach onehalf of the carrying capacity?

The Challenger Disaster After the Challenger disaster in 1986, a study was made of the 23 launches that preceded the fatal flight. A mathematical model was developed involving the relationship between the Fahrenheit temperature x around the O-rings and the number y of eroded or leaky primary O-rings. The model stated that 6 y = 1 + e -(5.085-0.l l56x ) where the number 6 indicates the 6 primary O-rings on the spacecraft. (a) What is the predicted number of eroded or leaky primary O-rings at a temperature of 1 00F? (b) What is the predicted number of eroded or leaky primary O-rings at a temperature of 60F? (c) What is the predicted number of eroded or leaky primary O-rings at a temperature of 30F? [i;TI (d) Graph the equation using a graphing utility. At what temperature is the predicted number of eroded or leaky O-rings I? 3? 5? Source: Linda Tappin, "Analyzing Data Relating to the Challenger Disaster," Mathematics Teacher, Vol. 87, No. 6, September 1994, pp. 423-426.

Biology A strain of E-coli Beu 397-recA441 is placed into a nutrient broth at 30Celsius and allowed to grow. The following data are collected. Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured. ------------ e (hours), x Population, y o 2.5 3.5 4.5 6 0.09 0.18 0.26 0.35 0.50 Source: Dr. Polly Lavery, Joliet Junior College (a) Draw a scatter diagram treating time as the independent variable. (b) Using a graphing utility, fit an exponential function to the data. ( c) Express the function found in part (b) in the form N(t) = Noekl. (d) Graph the exponential function found in part (b) or (c) on the scatter diagram. (e) Use the exponential function from part (b) or (c) to predict the population at x = 7 hours. (f) Use the exponential function from part (b) or ( c) to predict when the population will reach 0.75.

Biology A strain of E-coli SC18del-recA 718 is placed into a nutrient broth at 30Celsius and allowed to grow. The following data are collected. Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured. -.; ------------ TIme (hours), x Population, y 2.5 0.175 3.5 4.5 4.75 5.25 0.38 0.63 0.76 1 .20 Source: Dr. Polly Lavery, Joliet Junior College (a) Draw a scatter diagram treating time as the independent variable. (b) Using a graphing utility, fit an exponential function to the data. (c) Express the function found in part (b) in the form N(t) = Noekl. (d) Graph the exponential function found in part (b) or (c) on the scatter diagram. (e) Use the exponential function from part (b) or (c) to predict the population at x = 6 hours. (f) Use the exponential function from part (b) or (c) to predict when the population will reach 2.1.

Chemistry A chemist has a 100-gram sample of a radioactive material. He records the amount of radioactive material every week for 6 weeks and obtains the following data: Weight Week (in Grams) o 100.0 3 4 88.3 75.9 69.4 59.1 51.8 45.5 (a) Using a graphing utility, draw a scatter diagram with week as the independent variable. (b) Using a graphing utility, fit an exponential function to the data. (c) Express the function found in part (b) in the form A(t) = Aoekt (d) Graph the exponential function found in part (b) or (c) on the scatter diagram. (e) From the result found in part (b), determine tbe halflife of the radioactive material. (f) How much radioactive material will be left after 50 weeks? (g) When will there be 20 grams of radioactive material?

Chemistry A chemist has a lOaD-gram sample of a radioactive material. She records the amount of radioactive material remaining in tbe sample every day for a week and obtains the following data: Weight Day (in Grams) 0 1 000.0 897.1 2 802.5 3 71 9.8 4 651 .1 5 583.4 6 521 .7 7 468.3 (a) Using a graphing utility, draw a scatter diagram with day as the independent variable. (b) Using a graphing utility, fit an exponential function to the data. (c) Express the function found in part (b) in the form A(t) = Aoekl. (d) Grapb the exponential function found in part (b) or (c) on the scatter diagram. (e) From the result found in part (b), find the half-life of the radioactive material. (I') How much radioactive material will be left after 20 days? (g) When will there be 200 grams of radioactive material?

Cigarette Production The following data represent the number of cigarettes (in billions) produced in the United States by year. Cigarette Production Year (in billions of pieces) 1995 747 1 998 680 1999 607 2000 565 2001 562 2002 532 2003 499 2004 493 Source: Slalislical A bslraCI of the United Stales, 2006 (a) Let t = the number of years since 1995. Using a graphing utility, draw a scatter diagram of the data using I as tbe independent variable and number of cigarettes as the dependent variable. (b) Using a graphing utility, fit an exponential function to the data. (c) Express tbe function found in part (b) in the form A(t) = Aoekl. (d) Graph the exponential function found in part (b) or (c) on the scatter diagram. (e) Use the exponential function from part (b) or (c) to predict the number of cigarettes that will be produced in the United States in 2010. (f) Use the exponential function from part (b) or (c) to predict when the number of cigarettes produced in the United States will decrease to 230 billion.

Cigarette Exports The following data represent the number of cigarettes (in billions) exported from the Unjted States by year. Cigarette Exports (in billions of pieces) 1 995 231 .1 1 998 201 .3 1 999 151.4 2000 147.9 2001 133.9 2002 1 27.4 2003 121.5 2004 118.7 Source: Slalislical A bstract of lhe United Stales; 2006 (a) Let t = the number of years since 1995. Using a graphing utility, draw a scatter diagram of the data using I as the independent variable and number of cigarettes as the dependent variable. (b) Using a grapbing utility, fit an exponential function to the data. (c) Express the function found in part (b) in the form A (t) = Aoek '.(d) Graph the exponential function found in part (b) or (c) on the scatter diagram. (e) Use the exponential function from part (b) or (c) to predict the number of cigarettes that will be exported from the United States in 2010. (f) Use the exponential function from part (b) or (c) to predict when the number of cigarettes exported from the United States will decrease to 50 billion.

Economics and Marketing The following data represent the price and quantity demanded in 2005 for IBM personal computers. Price Quantity (S/Computer) Demanded 2300 1 52 2000 1 59 1 700 1 64 1 500 171 1 300 1 76 1 200 1 80 1 000 1 89 (a) Using a graphing utility, draw a scatter diagram of the data with price as the dependent variable. (b) Using a graphing utility, fit a logarithmic function to the data. (c) Using a graphing utility, draw the logarithmic function found in part (b) on the scatter diagram. (d) Use the function found in part (b) to predict the number of IBM personal computers that will be demanded if the price is $1650.

Economics and Marketing The following data represent the price and quantity supplied in 2005 for IBM personal com pu ters. Price Quantity (S/Computer) Supplied 2300 1 80 2000 1 73 1 700 1 60 1 500 1 50 1 300 1 37 1 200 1 30 1 000 113 (a) Using a graphing utility, draw a scatter diagram of the data with price as the dependent variable. (b) Using a graphing utility, fit a logarithmic function to the data. (c) Using a graphing utility, draw the logarithmic function found in part (b) on the scatter diagram. (d) Use the function found in part (b) to predict the number of IBM personal computers that will be supplied if the price is $1650.

Population Model The following data represent the population of the United States. An ecologist is interested in finding a function that describes the population of the United States. Year Population 1900 76,212,168 1910 92,228,496 1 920 1 06,021,537 1930 1 23,202,624 1 940 1 32,164,569 1 950 1 51,325,798 1 960 1 79,323,1 75 1 970 203,302,031 1 980 226,542,203 1 990 248,709,873 2000 281,421,906 Source: U.S. Census Bureau (a) Using a graphing utility, draw a scatter diagram of the data using the year as the independent variable and population as the dependent variable. (b) Using a graphing utility, fit a logistic function to the data. (c) Using a graphing utility, draw the function found in part (b) on the scatter diagram. (d) Based on the function found in part (b), what is the carrying capacity of the United States? (e) Use the function found in part (b) to predict the population of the United States in 2004. (f) When will the United States population be 300,000,000? (g) Compare actual U.S. Census figures to the prediction found in part (e). Discuss any differences.

Population Model The following data represent the world population. An ecologist is interested in finding a function that describes the world population. Population Year (in Billions) 1 993 5.531 1 994 5.61 1 1 995 5.691 1 996 5.769 1 997 5.847 1998 5.925 1 999 6.003 2000 6.080 2001 6.1 57 Source: U.S. Census B ureau (a) Using a graphing utility, draw a scatter diagram of the data using year as the independent variable and population as the dependent variable. (b) Using a graphing utility, fit a logistic function to the data. (c) Using a graphing utility, draw the function found in part (b) on the scatter diagram. (d) Based on the function found in part (b), what is the carrying capacity of the world? (e) Use the function found in part (b) to predict the population of the world in 2004. (f) When will world population be 7 billion? (g) Compare actual U.S. Census figures to the prediction found in part (e). Discuss any differences.

Cable Subscribers The following data represent the number of basic cable TV subscribers in the United States. A market !:""::1 r.. '?' ... __ Subscribers 1\1 Year (1,000) 1975 (t= 5) 9,800 1 980 (t= 10) 17,500 1 985 ( t = 1 5) 35,440 1990 (t= 20) 50,520 1 992 (t= 22) 54,300 1 994 (t= 24) 58,373 1 996 (t= 26) 62,300 1 998 (t= 28) 64,650 2000 (t= 30) 66,250 2002 (t= 32) 66,472 2004 (t= 34) 65,853 Source: Statistical Abstract of the United State,; 2006 CHAPTER REVIEW researcher believes that external factors, such as satellite TV, have affected the growth of cable subscribers. She is interested in finding a function that can be used to describe the number of cable TV subscribers in the United States. (a) Using a graphing utility, draw a scatter diagram of the data using the number of years after 1970, t, as the independent variable and number of subscribers as the dependent variable. (b) Using a graphing utility, fit a logistic function to the data. (c) Using a graphing utility, draw the function found in part (b) on the scatter diagram. (d) Based on the function found in part (b), what is the carrying capacity of the cable TV market in the United States? (e) Use the function found in part (b) to predict the number of cable TV subscribers in the United States in 2015.

Cell Phone Users Refer to the data in Table 9 (p. 488). (a) Using a graphing utility, fit a logistic function to the data. (b) Graph the logistic function found in part (b) on a scatter diagram of the data. (c) What is the predicted carrying capacity of U.S. cell phone subscribers? (d) Use the function found in part (b) to predict the number of U.S. cell phone subscribers at the end of 2009. (e) Compare the answer to part (d) above with the answer to Example 1, part (e). How do you explain the different predictions?

In Problems 1-6, for the given functions f and g find: (a) (f 0 g)(2) (b) (g 0 f)( -2) (c) (f 0 f)(4) (d) (g 0 g)( -1) f(x) = 3x - 5; g(x) = 1 - 2x2

In Problems 1-6, for the given functions f and g find: (a) (f 0 g)(2) (b) (g 0 f)( -2) (c) (f 0 f)(4) (d) (g 0 g)( -1) f(x) = 4 - x; g(x) = 1 + x2

In Problems 1-6, for the given functions f and g find: (a) (f 0 g)(2) (b) (g 0 f)( -2) (c) (f 0 f)(4) (d) (g 0 g)( -1) f(x) = Vx+2; g(x) = 2X2 + 1

In Problems 1-6, for the given functions f and g find: (a) (f 0 g)(2) (b) (g 0 f)( -2) (c) (f 0 f)(4) (d) (g 0 g)( -1) f(x) = 1 - 3x2; g(x) =

In Problems 1-6, for the given functions f and g find: (a) (f 0 g)(2) (b) (g 0 f)( -2) (c) (f 0 f)(4) (d) (g 0 g)( -1) f(x) = eX; g(x) = 3x - 2

In Problems 1-6, for the given functions f and g find: (a) (f 0 g)(2) (b) (g 0 f)( -2) (c) (f 0 f)(4) (d) (g 0 g)( -1) f(x) = 2 ; g(x) = 3x 1 + 2x

In Problems 7-12, find fog, g 0 f, f 0 f, and gog for each pair of functions. State the domain of each composite function. f(x) = 2 - x; g(x) = 3x + 1

In Problems 7-12, find fog, g 0 f, f 0 f, and gog for each pair of functions. State the domain of each composite function. f(x) = 2x - 1; g(x) = 2x + 1

In Problems 7-12, find fog, g 0 f, f 0 f, and gog for each pair of functions. State the domain of each composite function. f(x) = 3x2 + X + 1; g(x) = 13xl

In Problems 7-12, find fog, g 0 f, f 0 f, and gog for each pair of functions. State the domain of each composite function. f(x) = V3x; g(x) = 1 + x + x2

In Problems 7-12, find fog, g 0 f, f 0 f, and gog for each pair of functions. State the domain of each composite function. f(x) = --; g(x) = - x -I x

In Problems 7-12, find fog, g 0 f, f 0 f, and gog for each pair of functions. State the domain of each composite function. f(x) =; g(x) = l x

In Problems 13 and 14, (a) verify that the function is one-to-one, and (b) find the inverse of the given function. {(I, 2), (3,5), (5,8), (6, 10)}

In Problems 13 and 14, (a) verify that the function is one-to-one, and (b) find the inverse of the given function. {(-1,4), (0,2), (1,5), (3, 7)}

In Problems 15 and 16, state why the graph of the function is one-to-one. Then draw the graph of the inverse function f-I. For COI1Venience (and as a hint), the graph of y = x is also given. y 4 y=x (3,3) -4 (-1, -32.4

In Problems 15 and 16, state why the graph of the function is one-to-one. Then draw the graph of the inverse function f-I. For COI1Venience (and as a hint), the graph of y = x is also given. y 4 -4 y=x 4 x \ (, -1)

In Problems 17-22, the function f is one-to-one. Find the inverse of each function and check your answer. Find the domain and the range of f and F-1. f(x) =--5x - 2

In Problems 17-22, the function f is one-to-one. Find the inverse of each function and check your answer. Find the domain and the range of f and F-1. f(x) =--3 + x

In Problems 17-22, the function f is one-to-one. Find the inverse of each function and check your answer. Find the domain and the range of f and F-1. f(x) =-x - 1

In Problems 17-22, the function f is one-to-one. Find the inverse of each function and check your answer. Find the domain and the range of f and F-1. f(x) = V

In Problems 17-22, the function f is one-to-one. Find the inverse of each function and check your answer. Find the domain and the range of f and F-1. f(x) = 1/3 X

In Problems 17-22, the function f is one-to-one. Find the inverse of each function and check your answer. Find the domain and the range of f and F-1. f(x) = x l/3 + 1

In Problems 23 and 24, f(x) = 3"' and g(x) = IOg3 x. Evaluate: (a) f(4) (b) g(9) (c) f( - 2) (d) g ( 2)

In Problems 23 and 24, f(x) = 3"' and g(x) = IOg3 x. Evaluate: (a) f(1) (b) g(81) (c) f( - 4) (d) gC!3)

In Problems 25 and 26, convert each exponential expression to an equivalent expression involving a logarithm. 5 2 = Z

In Problems 25 and 26, convert each exponential expression to an equivalent expression involving a logarithm. as = m

In Problems 27 and 28, convert each logarithmic expression to an equivalent expression involving an exponent. logs u = 13

In Problems 27 and 28, convert each logarithmic expression to an equivalent expression involving an exponent. log" 4 = 3

In Problems 29-32, find the domain of each logarithmic fimction. f(x) = log(3x - 2)

In Problems 29-32, find the domain of each logarithmic fimction. F(x) = logs (2x + 1)

In Problems 29-32, find the domain of each logarithmic fimction. H(x) = log2 (x2 - 3x + 2)

In Problems 29-32, find the domain of each logarithmic fimction. F(x) = In (x2 - 9)

In Problems 33-38, evaluate each expression. Do not use a calculator. IOg2( )

In Problems 33-38, evaluate each expression. Do not use a calculator. log381

In Problems 33-38, evaluate each expression. Do not use a calculator. In eV

In Problems 33-38, evaluate each expression. Do not use a calculator. e 1nO.1

In Problems 33-38, evaluate each expression. Do not use a calculator. 2 log2 0.4

In Problems 33-38, evaluate each expression. Do not use a calculator. log2 2

In Problems 39-44, write each expression as the sum and/or difference of logarithms. Express powers as factors. IOg3( U: 2 } u > 0, v> 0, w >

In Problems 39-44, write each expression as the sum and/or difference of logarithms. Express powers as factors. log2 (a 2v'bt, a> 0, b >

In Problems 39-44, write each expression as the sum and/or difference of logarithms. Express powers as factors. log(x 2y':0+1) , x>

In Problems 39-44, write each expression as the sum and/or difference of logarithms. Express powers as factors. log5(x2 + 2x + 1), x > 0

In Problems 39-44, write each expression as the sum and/or difference of logarithms. Express powers as factors. n( x) , x> 3

In Problems 39-44, write each expression as the sum and/or difference of logarithms. Express powers as factors. n 2 ' x> 2 x - 3x + 2

In Problems 45-50, write each expression as a single logarithm. 3 IOg4 x- + "2 log 4

In Problems 45-50, write each expression as a single logarithm. -2 log3 (1) + 1log3 vx

In Problems 45-50, write each expression as a single logarithm. InC: 1 ) + InC: 1 ) - In (x2 - 1)

In Problems 45-50, write each expression as a single logarithm. log (x2 - 9) - log(x2 + 7x + 12)

In Problems 45-50, write each expression as a single logarithm. 2 1og2 + 3 10g x - "2 [log(x + 3) + log(x - 2)]

In Problems 45-50, write each expression as a single logarithm. 1 1 1 "2 ln (x2 + 1) - 4 1n "2 - "2 [ln (x - 4) + In x]

In Problems 51 and 52, use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. IOg419

In Problems 51 and 52, use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. IOg2 21

In Problems 53 and 54, graph each function using a graphing utility and the Change-of-Base Formula. y = log3 x

In Problems 53 and 54, graph each function using a graphing utility and the Change-of-Base Formula. Y = log7 X

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = y3

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = -2.1 + 3

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = (3-X)

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = 1 + rx

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = 1 - e-X

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = 3e x-2

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = 2 ln(x + 3)

In Problems 55-62, use the given function f to: (a) Find the domain of f (b) Graph f (c) From the graph, determine the range and any asymptotes of f (d) Find the inverse of f (e) User f1 to find the range of f (f) Graph f-1. f(x) = 3 + In(2x)

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 4 1-2 .\ = 2

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 8 6+3 .1 = 4

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 3 x2+x = V3

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 66 4 x-x' = . 2

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. logx 64 = -3

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. ogyz x = -6

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Sol = 3"'+2

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 5 x+2 = 7x- 2

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 92x = 273x-2

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 25 2.1 = 5 .\2-12

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log3=2

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2x+1 8 -x = 4

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 8 = 4.12 2 5 .1

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 2.1 5 = lOx

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log6 (x+3) + log6 (x+4) = 1

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. log(7x - 12) = 2 log x

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. e 1 -x = 5

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. el - 2x = 4

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 9.1 + 4 3 x - 3 = 0

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. 4.1 - 14,4-.1 = 5

Suppose that f(x) = log2(x - 2) + 1. (a) Graph f (b) What isf(6)? What point is on the graph off? (c) Solve f(x) = 4. What point is on the graph off? (d) Based on the graph drawn in part (a), solve f(x) > O. (e) Find r 1 (x) . Graph r 1 on the same Cartesian plane as f

Suppose that f(x) = lOg3(X + 1) - 4. (a) Graphf (b) What is f(8)? What point is on the graph of f? (c) Solve f(x) = -3. What point is on the graph off? (d) Based on the graph drawn in part (a), solve f(x) < O. (e) Find r1 (x). Graph r1 on the same Cartesian plane as f

In Problems 85 and 86, use the foLLowing result: If x is the atmospheric pressure (measured in miLLimeters of mercury), then the formula for the altitude hex) (measured in meters above sea level) is hex) = (30T + 8000) 10g( :0) where T is the temperature (in degrees Celsius) and Po is the atmospheric pressure at sea level, which is approximately 760 miLLimeters of mercury. Finding the Altitude of an Airplane At what height is a Piper Cub whose instruments record an outside temperature of OC and a barometric pressure of 300 millimeters of mercury?

In Problems 85 and 86, use the foLLowing result: If x is the atmospheric pressure (measured in miLLimeters of mercury), then the formula for the altitude hex) (measured in meters above sea level) is hex) = (30T + 8000) 10g( :0) where T is the temperature (in degrees Celsius) and Po is the atmospheric pressure at sea level, which is approximately 760 miLLimeters of mercury. Finding the Height of a Mountain How high is a mountain if instruments placed on its peak record a temperature of SoC and a barometric pressure of 500 millimeters of mercury?

Amplifying Sound An amplifier's power output P (in watts) is related to its decibel voltage gain d by the formula P = 25eo. 1d. (a) Find the power output for a decibel voltage gain of 4 decibels. (b) For a power output of 50 watts, what is the decibel voltage gain?

Limiting Magnitude of a Telescope A telescope is limited in its usefulness by the brightness of the star that it is aimed at and by the diameter of its lens. One measure of a star's brightness is its magnitude; the dimmer the star, the larger its magnitude. A formula for the limiting magnitude L of a telescope,that is, the magnitude of the dimmest star that it can be used to view, is given by L = 9 + 5.1 log d where d is the diameter (in inches) of the lens. (a) What is the limiting magnitude of a 3.5-inch telescope? (b) What diameter is required to view a star of magnitude 14?

Salvage Value The number of years n for a piece of machinery to depreciate to a known salvage value can be found using the formula log s - log i 11.= 10g(1 - d) where s is the salvage value of the machinery, i is its initial value, and d is the annual rate of depreciation. (a) How many years will it take for a piece of machinery to decline in value from $90,000 to $10,000 if the annual rate of depreciation is 0.20 (20%)? (b) How many years will it take for a piece of machinery to lose half of its value if the annual rate of depreciation is 15%?

Funding a College Education A child's grandparents purchase a $10,000 bond fund that matures in 18 years to be used for her college education. The bond fund pays 4% interest compounded semiannually. How much will the bond fund be worth at maturity? What is the effective rate of interest? How long will it take the bond to double in value under these terms?

Funding a College Education A child's grandparents wish to purchase a bond that matures in 18 years to be used for her college education. The bond pays 4% interest compounded semiannually. How much should they pay so that the bond will be worth $85,000 at maturity?

Funding an IRA First Colonial Bankshares Corporation advertised the following IRA investment plans. Target IRA Plans For each $5000 Maturity Value Desired Deposit: $620.17 $1045.02 $1760.92 $2967.26 At a Term of: 20 Years 15 Years 10 Years 5 Years (a) Assuming continuous compounding, what annual rate of interest did they offer? (b) First Colonial Bankshares claims that $4000 invested today will have a value of over $32,000 in 20 years. Use the answer found in part (a) to find the actual value of $4000 in 20 years. Assume continuous compounding.

Estimating the Date That a Prehistoric Man Died The bones of a prehistoric man found in the desert of New Mexico contain approximately 5% of the original amount of carbon 14. If the half-life of carbon 14 is 5600 years, approximately how long ago did the man die?

Temperature of a Skillet A skillet is removed from an oven whose temperature is 450F and placed in a room whose temperature is 70F After 5 minutes, the temperature of the skillet is 400F. How long will it be until its temperature is 150F?

World Population The annual growth rate of the world's population in 2005 was k = 1.15% = 0.0115. The popUlation of the world in 2005 was 6,451,058,790. Letting t = 0 represent 2005, use the uninhibited growth model to predict the world's population in the year 2015. Source: u.s. Census Bureau

Radioactive Decay The half-life of radioactive cobalt is 5.27 years. If 100 grams of radioactive cobalt is present now, how much will be present in 20 years? In 40 years?

Federal Deficit In fiscal year 2005, the federal deficit was $319 billion. At that time, 10-year treasury notes were paying 4.25% interest per annum. If the federal government financed this deficit through lO-year notes, how much would if have to pay back in 2015? Source: u.s. Treasury Department

Logistic Growth The logistic growth model 0.8 pet) = 1 + 1.67e - O. 161 represents the proportion of new cars with a global positioning system (GPS). Let t = 0 represent 2006, t = 1 represent 2007, and so on. (a) What proportion of new cars in 2006 had a GPS? (b) Determine the maximum proportion of new cars that have a GPS. O J: (c) Using a graphing utility, graph pet) . (d) When will 75% of new cars have a GPS?

CBL Experiment The following data were collected by placing a temperature probe in a portable heater, removing the probe, and then recording temperature over time. Time (sec.) Temperature (OF) 0 165.07 164.77 2 163.99 3 163.22 4 162.82 5 161.96 6 161.20 7 160.45 8 159.35 9 158.61 10 157.89 11 156.83 12 156.11 13 155.08 14 154.40 15 153.72 According to Newton's Law of Cooling, these data should follow an exponential model. (a) Using a graphing utility, draw a scatter diagram for the data. (b) Using a graphing utility, fit an exponential function to the data. 500 CHAPTER 6 Exponential and Logarithmic Functions ( c) Graph the exponential function found in part (b) on the scatter diagram. (d) Predict how long it will take for the probe to reach a temperature of llOF.

Wind Chill Factor The following data represent the wind speed (mph) and wind chill factor at an air temperature of 15F. Wind Speed (mph) Wind Chill Factor (OF) 5 7 10 3 15 0 20 -2 25 -4 30 -5 35 -7 Source: U.S. National Weather Service (a) Using a graphing utility, draw a scatter diagram with wind speed as the independent variable. (b) Using a graphing utility, fit a logarithmic function to the data. (c) Using a graphing utility, draw the logarithmic function found in part (b) on the scatter diagram. (d) Use the function found in part (b) to predict the wind chill factor if the air temperature is 15F and the wind speed is 23 mph.

Spreading of a Disease Jack and Diane live in a small town "of 50 people. Unfortunately, both Jack and Diane have a cold.Those who come in contact with someone who has this cold will themselves catch the cold. The following data represent the number of people in the small town who have caught the cold after t days. J :. J. f Days. t Number of People with Cold. C 0 2 4 2 8 3 14 4 22 5 30 6 37 7 42 8 44 (a) Using a graphing utility, draw a scatter diagram of the data. Comment on the type of relation that appears to exist between the days and number of people with a cold. (b) Using a graphing utility, fit a logistic function to the data. ( c) Graph the function found in part (b) on the scatter diagram. (d) According to the function found in part (b), what is the maximum number of people who will catch the cold? In reality, what is the maximum number of people who could catch the cold? (e) Sometime between the second and third day, 10 people in the town had a cold. According to the model found in part (b), when did 10 people have a cold? (f) How long will it take for 46 people to catch the cold?