Solution: Make a rough sketch of the graph of each function. Do not use a calculator

(a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one?

(a) Suppose f is a one-to-one function with domain A and range B. How is the inverse function f 21 defined? What is the domain of f 21 ? What is the range of f 21 ? (b) If you are given a formula for f, how do you find a formula for f 21 ? (c) If you are given the graph of f, how do you find the graph of f 21 ?

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. fsxd 2x 2 3

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. fsxd x 4 2 16

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.tsxd 1 2 sin x

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. tsxd s 3 x

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.fstd is the height of a football t seconds after kickoff.

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.fstd is your height at age t.

Assume that f is a one-to-one function. (a) If fs6d 17, what is f 21 s17d? (b) If f 21 s3d 2, what is fs2d?

If f sxd x 5 1 x 3 1 x, find f 21 s3d and fs f 21 s2dd.

If tsxd 3 1 x 1 ex , find t21 s4d.

The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f 21 ? (c) What is the value of f 21 s2d? (d) Estimate the value of f 21 s0d.

The formula C 5 9 sF 2 32d, where F > 2459.67, expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

In the theory of relativity, the mass of a particle with speed  is m fsvd m0 s1 2 v 2 yc 2 where m0 is the rest mass of the particle and c is the speed of light in a vacuum. Find the inverse function of f and explain its meaning.

Find a formula for the inverse of the function. fsxd 1 1 s2 1 3x

Find a formula for the inverse of the function. fsxd 4x 2 1 2x 1 3

Find a formula for the inverse of the function. fsxd e 2x21

Find a formula for the inverse of the function. y x 2 2 x, x > 1 2

Find a formula for the inverse of the function.y lnsx 1 3d

Find a formula for the inverse of the function.y 1 2 e2x 1 1 e2x

Find an explicit formula for f 21 and use it to graph f 21 , f, and the line y x on the same screen. To check your work, see whether the graphs of f and f 21 are reflections about the line. fsxd s4x 1 3

Find an explicit formula for f 21 and use it to graph f 21 , f, and the line y x on the same screen. To check your work, see whether the graphs of f and f 21 are reflections about the line. fsxd 1 1 e2x

Use the given graph of f to sketch the graph of f 21 .

Use the given graph of f to sketch the graph of f 21 .

Let f sxd s1 2 x 2 , 0 < x < 1. (a) Find f 21 . How is it related to f ? (b) Identify the graph of f and explain your answer to part (a).

Let tsxd s 3 1 2 x 3 . (a) Find t21 . How is it related to t? (b) Graph t. How do you explain your answer to part (a)?

(a) How is the logarithmic function y logb x defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function y logb x if b . 1.

(a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.

Find the exact value of each expression. (a) log2 32 (b) log8 2

Find the exact value of each expression. (a) log5 1 125 (b) lns1ye 2 d

Find the exact value of each expression. (a) log10 40 1 log10 2.5 (b) log8 60 2 log8 3 2 log8 5

Find the exact value of each expression. (a) e2ln 2 (b) elnsln e3 d

Express the given quantity as a single logarithm. ln 10 1 2 ln 5

Express the given quantity as a single logarithm. ln b 1 2 ln c 2 3 ln d

Express the given quantity as a single logarithm. 3 lnsx 1 2d 3 1 1 2 fln x 2 lnsx 2 1 3x 1 2d 2 g

Use Formula 10 to evaluate each logarithm correct to six decimal places. (a) log5 10 (b) log3 57

Use Formula 10 to graph the given functions on a common screen. How are these graphs related? y log1.5 x, y ln x, y log10 x, y log50 x

Use Formula 10 to graph the given functions on a common screen. How are these graphs related? y ln x, y log10 x, y ex , y 10x

Suppose that the graph of y log2 x is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft?

Compare the functions f sxd x 0.1 and tsxd ln x by graphing both f and t in several viewing rectangles. When does the graph of f finally surpass the graph of t?

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3. (a) y log10sx 1 5d (b) y 2ln x

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3. (a) y lns2xd (b) y ln | x |

(a) What are the domain and range of f ? (b) What is the x-intercept of the graph of f ? (c) Sketch the graph of f. fsxd ln x 1 2

(a) What are the domain and range of f ? (b) What is the x-intercept of the graph of f ? (c) Sketch the graph of f. fsxd lnsx 2 1d 2 1

Solve each equation for x. (a) e724x 6 (b) lns3x 2 10d 2

Solve each equation for x. (a) lnsx 2 2 1d 3 (b) e 2x 2 3ex 1 2 0

Solve each equation for x. (a) 2x25 3 (b) ln x 1 lnsx 2 1d 1

Solve each equation for x. (a) lnsln xd 1 (b) e ax Ce bx, where a b

Solve each inequality for x. (a) ln x , 0 (b) ex . 5

Solve each inequality for x. (a) 1 , e 3x21 , 2 (b) 1 2 2 ln x , 3

(a) Find the domain of fsxd lnsex 2 3d. (b) Find f 21 and its domain.

a) What are the values of eln 300 and lnse 300d? (b) Use your calculator to evaluate eln 300 and lnse 300d. What do you notice? Can you explain why the calculator has trouble?

Graph the function fsxd sx 3 1 x 2 1 x 1 1 and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for f 21 sxd. (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

(a) If tsxd x 6 1 x 4 , x > 0, use a computer algebra system to find an expression for t 21 sxd. (b) Use the expression in part (a) to graph y tsxd, y x, and y t 21 sxd on the same screen.

If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n fstd 100 2ty3 . (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50,000?

When a camera flash goes off, the batteries immediately begin to recharge the flashs capacitor, which stores electric charge given by Qstd Q0s1 2 e 2tya d (The maximum charge capacity is Q0 and t is measured in seconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of capacity if a 2?

Find the exact value of each expression. . (a) cos21 s21d (b) sin21 s0.5d

Find the exact value of each expression. (a) tan21 s3 (b) arctans21d

Find the exact value of each expression.(a) csc21 s2 (b) arcsin 1

Find the exact value of each expression. (a) sin21 (21ys2 ) (b) cos21 (s3 y2)

Find the exact value of each expression. . (a) cot21 (2s3 ) (b) sec21 2

Find the exact value of each expression. (a) arcsinssins5y4dd (b) cos(2 sin21 ( 5 13))

Prove that cosssin21 xd s1 2 x 2 .

Simplify the expression. tanssin21 xd

Simplify the expression. sinstan21 xd

Simplify the expression. sins2 arccos xd

Graph the given functions on the same screen. How are these graphs related? y sin x, 2y2 < x < y2; y sin21 x; y x

Graph the given functions on the same screen. How are these graphs related? y tan x, 2y2 , x , y2; y tan21 x; y x

Find the domain and range of the function tsxd sin21 s3x 1 1d

(a) Graph the function fsxd sinssin21 xd and explain the appearance of the graph. (b) Graph the function tsxd sin21 ssin xd. How do you explain the appearance of this graph?

(a) If we shift a curve to the left, what happens to its reflection about the line y x? In view of this geometric principle, find an expression for the inverse of tsxd fsx 1 cd, where f is a one-to-one function. (b) Find an expression for the inverse of hsxd fscxd, where c 0.