Beginning and Intermediate Algebra

Margaret L. Lial , John Hornsby

Chapter 12

Inverse, Exponential, and Logarithmic Functions - all with Video Answers

Educators

Section 1

Inverse Functions

Problem 1

A study found that the trans fat content in fast-food products varied widely around the world, based on the type of frying oil used, as shown in the table. If the set of countries is the domain and the set of trans fat percentages is the range of the function consisting of the six pairs listed, is it a one-to-one function? Why or why not?

Melissa S.

Melissa S.

Numerade Educator

Problem 2

The table shows the number of uncontrolled hazardous waste sites in 2008 that require further investigation to determine whether remedies are needed under the Superfund program. The eight states listed are ranked in the top ten on the EPA's National Priority List.
If this correspondence is considered to be a function that pairs each state with its number of uncontrolled waste sites, is it one-to-one? If not, explain why.

Melissa S.

Numerade Educator

Problem 3

The road mileage between Denver, Colorado, and several selected U.S. cities is shown in the table. If we consider this as a function that pairs each city with a distance, is it a one-to-one function? How could we change the answer to this question by adding 1 mile to one of the distances shown?

Melissa S.

Numerade Educator

Problem 4

Suppose you consider the set of ordered pairs $(x, y)$ such that $x$ represents a person in your mathematics class and $y$ represents that person's father. Explain how this function might not be a one-to-one function.

Melissa S.

Numerade Educator

Problem 5

If a function is made up of ordered pairs in such a way that the same $y$-value appears in a correspondence with two different $x$-values, then
A. the function is one-to-one
B. the function is not one-to-one
C. its graph does not pass the vertical line test
D. it has an inverse function associated with it.

Melissa S.

Numerade Educator

Problem 6

Which equation defines a one-to-one function? Explain why the others are not, using specific examples.
A. $f(x)=x$
B. $f(x)=x^{2}$
C. $f(x)=|x|$
D. $f(x)=-x^{2}+2 x-1$

Melissa S.

Numerade Educator

Problem 7

Only one of the graphs illustrates a one-to-one function. Which one is it?
(Check your book for graph)

Melissa S.

Numerade Educator

Problem 8

If a function $f$ is one-to-one and the point $(p, q)$ lies on the graph of $f$ then which point must lie on the graph of $f^{-1} ?$
A. $(-p, q)$
B. $(-q,-p)$
C. $(p,-q)$
D. $(q, p)$

Melissa S.

Numerade Educator

Problem 9

If the function is one-to-one, find its inverse.
$\{(3,6),(2,10),(5,12)\}$

Melissa S.

Numerade Educator

Problem 10

If the function is one-to-one, find its inverse.
$\left\{(-1,3),(0,5),(5,0),\left(7,-\frac{1}{2}\right)\right\}$

Melissa S.

Numerade Educator

Problem 11

If the function is one-to-one, find its inverse.
$\{(-1,3),(2,7),(4,3),(5,8)\}$

Melissa S.

Numerade Educator

Problem 12

If the function is one-to-one, find its inverse.
$\{(-8,6),(-4,3),(0,6),(5,10)\}$

Melissa S.

Numerade Educator

Problem 13

If the function is one-to-one, find its inverse.
$f(x)=2 x+4$

Melissa S.

Numerade Educator

Problem 14

If the function is one-to-one, find its inverse.
$f(x)=3 x+1$

Melissa S.

Numerade Educator

Problem 15

If the function is one-to-one, find its inverse.
$g(x)=\sqrt{x-3}, \quad x \geq 3$

Melissa S.

Numerade Educator

Problem 16

If the function is one-to-one, find its inverse.
$g(x)=\sqrt{x+2}, \quad x \geq-2$

Melissa S.

Numerade Educator

Problem 17

If the function is one-to-one, find its inverse.
$f(x)=3 x^{2}+2$

Melissa S.

Numerade Educator

Problem 18

If the function is one-to-one, find its inverse.
$f(x)=4 x^{2}-1$

Melissa S.

Numerade Educator

Problem 19

If the function is one-to-one, find its inverse.
$f(x)=x^{3}-4$

Melissa S.

Numerade Educator

Problem 20

If the function is one-to-one, find its inverse.
$f(x)=x^{3}+5$

Melissa S.

Numerade Educator

Problem 21

Let $f(x)=2^{x} .$ We will see in the next section that this function is one-toone. Find each value, always working part (a) before part $(b)$.

(a) $f(3)$
(b) $f^{-1}(8)$

Melissa S.

Numerade Educator

Problem 22

Let $f(x)=2^{x} .$ We will see in the next section that this function is one-toone. Find each value, always working part (a) before part $(b)$.

(a) $f(4)$
(b) $f^{-1}(16)$

Melissa S.

Numerade Educator

Problem 23

Let $f(x)=2^{x} .$ We will see in the next section that this function is one-toone. Find each value, always working part (a) before part $(b)$.

(a) $f(0)$
(b) $f^{-1}(1)$

Melissa S.

Numerade Educator

Problem 24

Let $f(x)=2^{x} .$ We will see in the next section that this function is one-toone. Find each value, always working part (a) before part $(b)$.

(a) $f(-2)$
(b) $f^{-1}\left(\frac{1}{4}\right)$

Melissa S.

Numerade Educator

Problem 25

(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$)
(Check your book for graph)

Melissa S.

Numerade Educator

Problem 26

(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$)
(Check your book for graph)

Melissa S.

Numerade Educator

Problem 27

(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$)
(Check your book for graph)

Melissa S.

Numerade Educator

Problem 28

(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$)
(Check your book for graph)

Melissa S.

Numerade Educator

Problem 29

(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$)
(Check your book for graph)

Melissa S.

Numerade Educator

Problem 30

(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$)
(Check your book for graph)

Melissa S.

Numerade Educator

Problem 31

Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).
$f(x)=2 x-1$

Melissa S.

Numerade Educator

Problem 32

Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).
$f(x)=2 x+3$

Melissa S.

Numerade Educator

Problem 33

Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).
$g(x)=-4 x$

Melissa S.

Numerade Educator

Problem 34

Graph the function as a solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or curve).
$g(x)=-2 x$

Melissa S.

Numerade Educator

Problem 35

Complete the table so that graphing the function will be easier.
$f(x)=\sqrt{x}$
$x \geq 0$
$\begin{array}{c|c}{x} & {f(x)} \\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 4 & {}\end{array}$

Melissa S.

Numerade Educator

Problem 36

Complete the table so that graphing the function will be easier.
$f(x)=-\sqrt{x}$
$x \geq 0$
$\begin{array}{c|c}{x} & {f(x)} \\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 4 & {}\end{array}$

Melissa S.

Numerade Educator

Problem 37

Complete the table so that graphing the function will be easier.
$f(x)=x^{3}-2$
$\begin{array}{c|c}{x} & {f(x)} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2\end{array}$

Melissa S.

Numerade Educator

Problem 38

Complete the table so that graphing the function will be easier.
$f(x)=x^{3}+3$
$\begin{array}{c|c}{x} & {f(x)} \\ \hline-2 & {} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline 1 & {}\end{array}$

Melissa S.

Numerade Educator

Problem 39

Suppose that you are an agent for a detective agency. Today's function for your code is defined by $f(x)=4 x-5 .$ Find the rule for $f^{-1}$ algebraically.

Melissa S.

Numerade Educator

Problem 40

You receive the following coded message today. (Read across from left to right.)
$\begin{array}{lllllllllllll}{47} & {95} & {23} & {67} & {-1} & {59} & {27} & {31} & {51} & {23} & {7-1} & {43} & {779} & {43} & {-1} & {75} & {55} & {67}\end{array}$
$\begin{array}{llllllllllllll}{31} & {71} & {75} & {27} & {15} & {23} & {67} & {15} & {-1} & {75} & {15} & {71} & {75} & {75} & {27} & {31} & {51}\end{array}$
$\begin{array}{llllllllllllll}{23} & {71} & {31} & {51} & {715} & {71} & {43} & {31} & {7} & {15} & {11} & {367} & {15} & {-1} & {11}\end{array}$
Use the letter/number assignment described earlier to decode the message.

Melissa S.

Numerade Educator

Problem 41

Why is a one-to-one function essential in this encoding/decoding process?

Melissa S.

Numerade Educator

Problem 42

Use $f(x)=x^{3}+4$ to encode your name, using the letter/number assignment described earlier.

Melissa S.

Numerade Educator

Problem 43

Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.
$f(x)=2 x-7$

Melissa S.

Numerade Educator

Problem 44

Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.
$f(x)=-3 x+2$

Melissa S.

Numerade Educator

Problem 45

Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.
$f(x)=x^{3}+5$

Melissa S.

Numerade Educator

Problem 46

Function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window.
$f(x)=\sqrt[3]{x+2}$

Melissa S.

Numerade Educator

Problem 47

If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.
$f(3)$

Melissa S.

Numerade Educator

Problem 48

If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.
$f\left(\frac{1}{2}\right)$

Melissa S.

Numerade Educator

Problem 49

If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.
$f\left(-\frac{1}{2}\right)$

Melissa S.

Numerade Educator

Problem 50

If $f(x)=4^{x},$ find value indicated. Use a calculator, and give the answer to the nearest hundredth.
$f(2.73)$

Melissa S.

Numerade Educator