Solved: 2128. Finding inverse functions a. Find the

For b 7 0, what are the domain and range of f 1x2 = b x?

Give an example of a function that is one-to-one on the entire real number line.

Explain why a function that is not one-to-one on an interval I cannot have an inverse function on I.

Explain with pictures why 1a, b2 is on the graph of f whenever 1b, a2 is on the graph of f -1.

Sketch a function that is one-to-one and positive for x 0. Make a rough sketch of its inverse.

Express the inverse of f 1x2 = 3x - 4 in the form y = f -11x).

Explain the meaning of logb x.

How is the property b x + y = b xb y related to the property logb 1xy2 = logb x + logb y?

For b 7 0 with b _ 1, what are the domain and range of f 1x2 = logb x and why?

Express 25 using base e.

1114. One-to-one functions Find three intervals on which f is one-to-one, making each interval as large as possible. _3 _2 _1 1 2 3 y x y _ f (x)

1114. One-to-one functions Find four intervals on which f is one-to-one, making each interval as large as possible. _4 _2 2 4 y x y _ f (x)

1114. One-to-one functions Sketch a graph of a function that is one-to-one on the interval 1- _, 04 but is not one-to-one on 1- _, _).

1114. One-to-one functions Sketch a graph of a function that is one-to-one on the intervals 1- _, -24, and 3-2, _2 but is not one-to-one on 1- _, _).

1520. Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. f 1x2 = 3x + 4

1520. Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. f 1x2 = _ 2x + 1 _

1520. Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. f 1x2 = 1>1x - 52

1520. Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. f 1x2 = -16 - x22

1520. Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. f 1x2 = 1>x2

1520. Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. f 1x2 = x2 - 2x + 8 (Hint: Complete the square.)

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = 2x

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = 1 4 x + 1

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = 6 - 4x

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = 3x3

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = 3x + 5

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = x2 + 4, for x 0

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = 1x + 2, for x -2

2128. Finding inverse functions a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f -11x). b. Verify the relationships f 1 f -11x22 = x and f -11 f 1x22 = x. f 1x2 = 2>1x2 + 12, for x 0

Splitting up curves The unit circle x2 + y2 = 1 consists of four one-to-one functions, f11x2, f21x2, f31x2, and f41x2 (see figure). a. Find the domain and a formula for each function. b. Find the inverse of each function and write it as y = f -11x).

Splitting up curves The equation y4 = 4x2 is associated with four one-to-one functions f11x2, f21x2, f31x2, and f41x2 (see figure). a. Find the domain and a formula for each function. b. Find the inverse of each function and write it as y = f - 11x). .

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = 8 - 4x

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = 4x - 12

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = 1x, for x 0

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = 13 - x, for x 3

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = x4 + 4, for x 0

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = 6>1x2 - 92, for x 7 3

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = x2 - 2x + 6, for x 1 (Hint: Complete the square first.)

3138. Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f -1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. f 1x2 = -x2 - 4x - 3, for x -2 (Hint: Complete the square first.)

3940. Graphs of inverses Sketch the graph of the inverse function.

3940. Graphs of inverses Sketch the graph of the inverse function.

4146. Solving logarithmic equations Solve the following equations. log10 x = 3

4146. Solving logarithmic equations Solve the following equations. log5 x = -1

4146. Solving logarithmic equations Solve the following equations. log8 x = 13

4146. Solving logarithmic equations Solve the following equations. logb 125 = 3

4146. Solving logarithmic equations Solve the following equations. ln x = -1

4146. Solving logarithmic equations Solve the following equations. ln y = 3

4752. Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb x y

4752. Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb x2

4752. Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb xz

4752. Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb 1xy z

4752. Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb 1x 23 z

4752. Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. logb b2x5>2 1y

5356. Solving equations Solve the following equations. 7x = 21

5356. Solving equations Solve the following equations. 2x = 55

5356. Solving equations Solve the following equations. 33x - 4 = 15

5356. Solving equations Solve the following equations. 53x = 29

Using inverse relations One hundred grams of a particular radioactive substance decays according to the function m1t2 = 100 e-t>650, where t 7 0 measures time in years. When does the mass reach 50 grams?

Using inverse relations The population P of a small town grows according to the function P1t2 = 100 et>50, where t measures the number of years after 2010. How long does it take the population to double? .

5962. Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. log2 15

5962. Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. log3 30

5962. Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. log4 40

5962. Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. log6 60

6368. Changing bases Convert the following expressions to the indicated base. 2x using base e

6368. Changing bases Convert the following expressions to the indicated base. 3sin x using base e

6368. Changing bases Convert the following expressions to the indicated base. ln _ x _ using base 5

6368. Changing bases Convert the following expressions to the indicated base. log2 1x2 + 12 using base e

6368. Changing bases Convert the following expressions to the indicated base. a1>ln a using base e, for a 7 0 and a _ 1

6368. Changing bases Convert the following expressions to the indicated base. a1>log10 a using base 10, for a 7 0 and a _ 1

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If y = 3x, then x = 23 y. b. logb x logb y = logb x - logb y c. log5 46 = 4 log5 6 d. 2 = 10log10 2 e. 2 = ln 2e f. If f 1x2 = x2 + 1, then f -11x2 = 1>1x2 + 1). g. If f 1x2 = 1>x, then f -11x2 = 1>x.

Graphs of exponential functions The following figure shows the graphs of y = 2x, y = 3x, y = 2-x, and y = 3-x. Match each curve with the correct function.

Graphs of logarithmic functions The following figure shows the graphs of y = log2 x, y = log4 x, and y = log10 x. Match each curve with the correct function.

Graphs of modified exponential functions Without using a graphing utility, sketch the graph of y = 2x. Then on the same set of axes, sketch the graphs of y = 2-x, y = 2x - 1, y = 2x + 1, and y = 22x.

Graphs of modified logarithmic functions Without using a graphing utility, sketch the graph of y = log2 x. Then on the same set of axes, sketch the graphs of y = log2 1x - 12, y = log2 x2, y = 1log2 x22, and y = log2 x + 1.

Large intersection point Use any means to approximate the intersection point(s) of the graphs of f 1x2 = ex and g1x2 = x123. (Hint: Consider using logarithms.)

7578. Finding all inverses Find all the inverses associated with the following functions and state their domains. f 1x2 = 1x + 123

7578. Finding all inverses Find all the inverses associated with the following functions and state their domains. f 1x2 = 1x - 422

7578. Finding all inverses Find all the inverses associated with the following functions and state their domains. f 1x2 = 2>1x2 + 22

7578. Finding all inverses Find all the inverses associated with the following functions and state their domains. f 1x2 = 2x>1x + 22

Population model A culture of bacteria has a population of 150 cells when it is first observed. The population doubles every 12 hr, which means its population is governed by the function p1t2 = 150 # 2t>12, where t is the number of hours after the first observation. a. Verify that p102 = 150, as claimed. b. Show that the population doubles every 12 hr, as claimed. c. What is the population 4 days after the first observation? d. How long does it take the population to triple in size? e. How long does it take the population to reach 10,000?

Charging a capacitor A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function Q1t2 = a11 - e-t>c2, where t is measured in seconds, and a and c 7 0 are physical constants. The steady-state charge is the value that Q1t2 approaches as t becomes large. a. Graph the charge function for t 0 using a = 1 and c = 10. Find a graphing window that shows the full range of the function. b. Vary the value of a while holding c fixed. Describe the effect on the curve. How does the steady-state charge vary with a? c. Vary the value of c while holding a fixed. Describe the effect on the curve. How does the steady-state charge vary with c? d. Find a formula that gives the steady-state charge in terms of a and c.

Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft>s is given by h = f 1t2 = 64t - 16t2, where t is measured in seconds after the hit. a. Is this function one-to-one on the interval 0 t 4? b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = f -1 1h). c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = f -1 1h). d. At what time is the ball at a height of 30 ft on the way up? e. At what time is the ball at a height of 10 ft on the way down?

Velocity of a skydiver The velocity of a skydiver 1in m>s2 t seconds after jumping from a plane is v1t2 = 60011 - e-kt>602>k, where k 7 0 is a constant. The terminal velocity of the skydiver is the value that v1t2 approaches as t becomes large. Graph v with k = 11 and estimate the terminal velocity.

Reciprocal bases Assume that b 7 0 and b _ 1. Show that log1>b x = - logb x.

Proof of rule L1 Use the following steps to prove that logb xy = logb x + logb y. a. Let x = bp and y = bq. Solve these expressions for p and q, respectively. b. Use property E1 for exponents to express xy in terms of b, p, and q. c. Compute logb xy and simplify.

Proof of rule L2 Modify the proof outlined in Exercise 84 and use property E2 for exponents to prove that logb 1x>y2 = logb x - logb y.

Proof of rule L3 Use the following steps to prove that logb xz = z logb x. a. Let x = bp. Solve this expression for p. b. Use property E3 for exponents to express xz in terms of b and p. c. Compute logb xz and simplify. T T

Inverses of a quartic Consider the quartic polynomial y = f 1x2 = x4 - x2. a. Graph f and estimate the largest intervals on which it is oneto- one. The goal is to find the inverse function on each of these intervals. b. Make the substitution u = x2 to solve the equation y = f 1x2 for x in terms of y. Be sure you have included all possible solutions. c. Write each inverse function in the form y = f -11x2 for each of the intervals found in part (a).

Inverse of composite functions a. Let g1x2 = 2x + 3 and h1x2 = x3. Consider the composite function f 1x2 = g1h1x22. Find f -1 directly and then express it in terms of g-1 and h-1. b. Let g1x2 = x2 + 1 and h1x2 = 1x. Consider the composite function f 1x2 = g1h1x22. Find f -1 directly and then express it in terms of g-1 and h-1. c. Explain why if g and h are one-to-one, the inverse of f 1x2 = g1h1x22 exists.

8990. Inverses of (some) cubics Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic y = f 1x2 = x3 + ax. Find the inverse of the following cubics using the substitution (known as Vietas substitution) x = z - a>13z). Be sure to determine where the function is one-to-one. f 1x2 = x3 + 2x

8990. Inverses of (some) cubics Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic y = f 1x2 = x3 + ax. Find the inverse of the following cubics using the substitution (known as Vietas substitution) x = z - a>13z). Be sure to determine where the function is one-to-one. f 1x2 = x3 + 4x - 1

Nice property Prove that 1logb c21logc b2 = 1, for b 7 0, c 7 0, b _ 1, and c _ 1.