• Home
  • Textbooks
  • Intermediate Algebra : Concepts and Applications
  • Exponential Functions and Logarithmic Functions

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

Chapter 9

Exponential Functions and Logarithmic Functions - all with Video Answers

Educators

AG

Section 1

Composite Functions and Inverse Functions

00:53

Problem 1

Classify the following statements as either true or false.
The composition of two functions $f$ and $g$ is written $f \circ g$.

Caleb Wood
Caleb Wood
Numerade Educator
01:13

Problem 2

Classify the following statements as either true or false.
The notation $(f \circ g)(x)$ means $f(g(x))$.

Caleb Wood
Caleb Wood
Numerade Educator
01:34

Problem 3

Classify the following statements as either true or false.
If $f(x)=x^{2}$ and $g(x)=x+3,$ then $(g \circ f)(x)=$ $(x+3)^{2}$.

Caleb Wood
Caleb Wood
Numerade Educator
02:21

Problem 4

Classify the following statements as either true or false.
For any function $h,$ there is only one way to decompose the function as $h=f \circ g$.

Caleb Wood
Caleb Wood
Numerade Educator
02:02

Problem 5

Classify the following statements as either true or false.
The function $f$ is one-to-one if $f(1)=1$.

Caleb Wood
Caleb Wood
Numerade Educator
01:43

Problem 6

Classify the following statements as either true or false.
The $-1$ in $f^{-1}$ is an exponent.

Caleb Wood
Caleb Wood
Numerade Educator
02:39

Problem 7

Classify the following statements as either true or false.
The function $f$ is the inverse of $f^{-1}$.

Caleb Wood
Caleb Wood
Numerade Educator
02:16

Problem 8

Classify the following statements as either true or false.
If $g$ and $h$ are inverses of each other, then $(g \circ h)(x)=x$.

Caleb Wood
Caleb Wood
Numerade Educator
05:19

Problem 9

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=x^{2}+1 ; g(x)=x-3$

Caleb Wood
Caleb Wood
Numerade Educator
04:43

Problem 10

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=x+4 ; g(x)=x^{2}-5$

Caleb Wood
Caleb Wood
Numerade Educator
06:30

Problem 11

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=5 x+1 ; g(x)=2 x^{2}-7$

Caleb Wood
Caleb Wood
Numerade Educator
06:51

Problem 12

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=3 x^{2}+4 ; g(x)=4 x-1$

Caleb Wood
Caleb Wood
Numerade Educator
04:40

Problem 13

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=x+7 ; g(x)=1 / x^{2}$

Caleb Wood
Caleb Wood
Numerade Educator
04:23

Problem 14

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=1 / x^{2} ; g(x)=x+2$

Caleb Wood
Caleb Wood
Numerade Educator
04:08

Problem 15

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=\sqrt{x} ; g(x)=x+3$

Caleb Wood
Caleb Wood
Numerade Educator
04:37

Problem 16

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=10-x ; g(x)=\sqrt{x}$

Caleb Wood
Caleb Wood
Numerade Educator
04:47

Problem 17

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=\sqrt{4 x} ; g(x)=1 / x$

Caleb Wood
Caleb Wood
Numerade Educator
04:44

Problem 18

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=\sqrt{x+3} ; g(x)=13 / x$

Caleb Wood
Caleb Wood
Numerade Educator
04:48

Problem 19

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=x^{2}+4 ; g(x)=\sqrt{x-1}$

Caleb Wood
Caleb Wood
Numerade Educator
05:25

Problem 20

For pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;(\mathbf{d})(g \circ f)(x)$.

$f(x)=x^{2}+8 ; g(x)=\sqrt{x+17}$

Caleb Wood
Caleb Wood
Numerade Educator
02:40

Problem 21

Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x) .$ Answers may vary.
$h(x)=(3 x-5)^{4}$

Caleb Wood
Caleb Wood
Numerade Educator
02:12

Problem 22

Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x) .$ Answers may vary.
$h(x)=(2 x+7)^{3}$

Caleb Wood
Caleb Wood
Numerade Educator
01:28

Problem 23

Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x) .$ Answers may vary.
$h(x)=\sqrt{9 x+1}$

AG
Ankit Gupta
Numerade Educator
02:16

Problem 24

Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x) .$ Answers may vary.
$h(x)=\sqrt[3]{4 x-5}$

Caleb Wood
Caleb Wood
Numerade Educator
01:17

Problem 25

Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x) .$ Answers may vary.
$h(x)=\frac{6}{5 x-2}$

AG
Ankit Gupta
Numerade Educator
02:15

Problem 26

Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x) .$ Answers may vary.
$h(x)=\frac{3}{x}+4$

Caleb Wood
Caleb Wood
Numerade Educator
00:32

Problem 27

Determine whether each function is one-to-one.
$f(x)=-x$

James Kiss
James Kiss
Numerade Educator
01:48

Problem 28

Determine whether each function is one-to-one.
$f(x)=x+5$

AG
Ankit Gupta
Numerade Educator
00:49

Problem 29

Determine whether each function is one-to-one.
$f(x)=x^{2}+3$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:30

Problem 30

Determine whether each function is one-to-one.
$f(x)=3-x^{2}$

AG
Ankit Gupta
Numerade Educator
01:09

Problem 31

Determine whether each function is one-to-one.
(Check your book for graph)

AG
Ankit Gupta
Numerade Educator
01:46

Problem 32

Determine whether each function is one-to-one.
(Check your book for graph)

AG
Ankit Gupta
Numerade Educator
01:57

Problem 33

Determine whether each function is one-to-one.
(Check your book for graph)

AG
Ankit Gupta
Numerade Educator
02:02

Problem 34

Determine whether each function is one-to-one.
(Check your book for graph)

AG
Ankit Gupta
Numerade Educator
01:54

Problem 35

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=x+3$

AG
Ankit Gupta
Numerade Educator
02:31

Problem 36

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=x+2$

Caleb Wood
Caleb Wood
Numerade Educator
02:22

Problem 37

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=2 x$

Caleb Wood
Caleb Wood
Numerade Educator
02:08

Problem 38

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=3 x$

Caleb Wood
Caleb Wood
Numerade Educator
02:36

Problem 39

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$g(x)=3 x-1$

Caleb Wood
Caleb Wood
Numerade Educator
02:13

Problem 40

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$g(x)=2 x-3$

AG
Ankit Gupta
Numerade Educator
02:34

Problem 41

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\frac{1}{2} x+1$

Caleb Wood
Caleb Wood
Numerade Educator
02:45

Problem 42

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\frac{1}{3} x+2$

Caleb Wood
Caleb Wood
Numerade Educator
02:22

Problem 43

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$g(x)=x^{2}+5$

Caleb Wood
Caleb Wood
Numerade Educator
02:17

Problem 44

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$g(x)=x^{2}-4$

Caleb Wood
Caleb Wood
Numerade Educator
02:29

Problem 45

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$h(x)=-10-x$

Caleb Wood
Caleb Wood
Numerade Educator
02:40

Problem 46

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$h(x)=7-x$

Caleb Wood
Caleb Wood
Numerade Educator
02:50

Problem 47

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\frac{1}{x}$

Caleb Wood
Caleb Wood
Numerade Educator
01:35

Problem 48

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\frac{4}{x}$

AG
Ankit Gupta
Numerade Educator
01:44

Problem 49

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$g(x)=1$

AG
Ankit Gupta
Numerade Educator
00:58

Problem 50

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$h(x)=8$

AG
Ankit Gupta
Numerade Educator
02:52

Problem 51

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\frac{2 x+1}{3}$

Caleb Wood
Caleb Wood
Numerade Educator
02:54

Problem 52

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\frac{3 x+2}{5}$

Caleb Wood
Caleb Wood
Numerade Educator
02:03

Problem 53

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=x^{3}+5$

AG
Ankit Gupta
Numerade Educator
02:35

Problem 54

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=x^{3}-4$

AG
Ankit Gupta
Numerade Educator
02:43

Problem 55

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$g(x)=(x-2)^{3}$

Caleb Wood
Caleb Wood
Numerade Educator
03:13

Problem 56

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$g(x)=(x+7)^{3}$

Caleb Wood
Caleb Wood
Numerade Educator
02:17

Problem 57

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\sqrt{x}$

Caleb Wood
Caleb Wood
Numerade Educator
02:36

Problem 58

For each function, (a) determine whether it is one-to-one; (b) if it is one-to-one, find a formula for the inverse.
$f(x)=\sqrt{x-1}$

Caleb Wood
Caleb Wood
Numerade Educator
08:51

Problem 59

A size- 6 dress in the United States is size 36 in Italy. A function that converts dress sizes in the United States to those in Italy is $f(x)=2(x+12)$.
a) Find the dress sizes in Italy that correspond to sizes $8,10,14,$ and 18 in the United States.
b) Does $f$ have an inverse that is a function? If so, find a formula for the inverse.
c) Use the inverse function to find dress sizes in the United States that correspond to sizes 40, $44,52,$ and 60 in Italy.

AG
Ankit Gupta
Numerade Educator
08:50

Problem 60

A size-6 dress in the United States is size 38 in France. A function that converts dress sizes in the United States to those in France is $f(x)=x+32$.
a) Find the dress sizes in France that correspond to sizes $8,10,14,$ and 18 in the United States.
b) Does $f$ have an inverse that is a function? If so, find a formula for the inverse.
c) Use the inverse function to find dress sizes in the United States that correspond to sizes 40
$42,46,$ and 50 in France.

AG
Ankit Gupta
Numerade Educator
02:49

Problem 61

Graph function and its inverse using the same set of axes.
$f(x)=\frac{2}{3} x+4$

Caleb Wood
Caleb Wood
Numerade Educator
02:26

Problem 62

Graph function and its inverse using the same set of axes.
$g(x)=\frac{1}{4} x+2$

Caleb Wood
Caleb Wood
Numerade Educator
02:17

Problem 63

Graph function and its inverse using the same set of axes.
$f(x)=x^{3}+1$

Caleb Wood
Caleb Wood
Numerade Educator
02:58

Problem 64

Graph function and its inverse using the same set of axes.
$f(x)=x^{3}-1$

Caleb Wood
Caleb Wood
Numerade Educator
02:27

Problem 65

Graph function and its inverse using the same set of axes.
$g(x)=\frac{1}{2} x^{3}$

Caleb Wood
Caleb Wood
Numerade Educator
02:34

Problem 66

Graph function and its inverse using the same set of axes.
$g(x)=\frac{1}{3} x^{3}$

Caleb Wood
Caleb Wood
Numerade Educator
02:13

Problem 67

Graph function and its inverse using the same set of axes.
$F(x)=-\sqrt{x}$

AG
Ankit Gupta
Numerade Educator
03:50

Problem 68

Graph function and its inverse using the same set of axes.
$f(x)=\sqrt{x}$

Caleb Wood
Caleb Wood
Numerade Educator
03:30

Problem 69

Graph function and its inverse using the same set of axes.
$f(x)=-x^{2}, x \geq 0$

Caleb Wood
Caleb Wood
Numerade Educator
04:47

Problem 70

Graph function and its inverse using the same set of axes.
$f(x)=x^{2}-1, x \leq 0$

Caleb Wood
Caleb Wood
Numerade Educator
01:07

Problem 71

Let $f(x)=\sqrt[3]{x}-4 .$ Use composition of inverse functions to show that
$$f^{-1}(x)=x^{3}+4$$

AG
Ankit Gupta
Numerade Educator
05:19

Problem 72

Let $f(x)=3 /(x+2) .$ Use composition of inverse functions to show that
$$f^{-1}(x)=\frac{3}{x}-2$$

AG
Ankit Gupta
Numerade Educator
01:11

Problem 73

Let $f(x)=(1-x) / x .$ Use composition of inverse functions to show that
$$f^{-1}(x)=\frac{1}{x+1}$$

AG
Ankit Gupta
Numerade Educator
04:49

Problem 74

Let $f(x)=x^{3}-5 .$ Use composition of inverse functions to show that
$$f^{-1}(x)=\sqrt[3]{x+5}$$

AG
Ankit Gupta
Numerade Educator
02:58

Problem 75

Is there a one-to-one relationship between items in a store and the price of each of those items? Why or why not?

Caleb Wood
Caleb Wood
Numerade Educator
01:44

Problem 76

Mathematicians usually try to select "logical" words when forming definitions. Does the term "one-to-one" seem logical? Why or why not?

AG
Ankit Gupta
Numerade Educator
01:27

Problem 77

Simplify.
$t^{1 / 5} t^{2 / 3}$

AG
Ankit Gupta
Numerade Educator
01:32

Problem 78

Simplify.
$\sqrt[3]{40 a^{5} b^{12}}$

AG
Ankit Gupta
Numerade Educator
02:17

Problem 79

Simplify.
$\left(-3 x^{-6} y^{4}\right)^{-2}$

AG
Ankit Gupta
Numerade Educator
01:26

Problem 80

Simplify.
$i^{43}$

AG
Ankit Gupta
Numerade Educator
01:51

Problem 81

Simplify.
$3^{3}+2^{2}-(32 \div 4-16 \div 8)$

AG
Ankit Gupta
Numerade Educator
01:21

Problem 82

Simplify.
$\left(1.5 \times 10^{-3}\right)\left(4.2 \times 10^{-12}\right)$

AG
Ankit Gupta
Numerade Educator
02:40

Problem 83

The function $V(t)=750(1.2)^{t}$ is used to predict the value $V(t)$ of a certain rare stamp $t$ years after $2016 .$ Do not calculate $V^{-1}(t),$ but explain how $V^{-1}$ could be used.

AG
Ankit Gupta
Numerade Educator
00:59

Problem 84

An organization determines that the cost per person $C(x),$ in dollars, of chartering a bus with $x$ passengers is given by
$$C(x)=\frac{100+5 x}{x}$$
Determine $C^{-1}(x)$ and explain how this inverse function could be used.

AG
Ankit Gupta
Numerade Educator
00:59

Problem 85

Graph the inverse of $f$.
(Check your book for graph)

AG
Ankit Gupta
Numerade Educator
00:51

Problem 86

Graph the inverse of $f$.
(Check your book for graph)

AG
Ankit Gupta
Numerade Educator
01:48

Problem 87

Use the information in Exercises 59 and 60 to find a function for the dress size in France that corresponds to a size $x$ dress in Italy.

AG
Ankit Gupta
Numerade Educator
01:58

Problem 88

Use the information in Exercises 59 and 60 to find a function for the dress size in Italy that corresponds to a size $x$ dress in France.

AG
Ankit Gupta
Numerade Educator
00:54

Problem 89

What relationship exists between the answers to Exercises 87 and $88 ?$ Explain how you determined this.

AG
Ankit Gupta
Numerade Educator
05:33

Problem 90

Show that function composition is associative by showing that $((f \circ g) \circ h)(x)=(f \circ(g \circ h))(x)$

Caleb Wood
Caleb Wood
Numerade Educator
05:33

Problem 91

Show that function composition is associative by showing that $((f \circ g) \circ h)(x)=(f \circ(g \circ h))(x)$.

Caleb Wood
Caleb Wood
Numerade Educator
03:34

Problem 92

Determine whether or not the given pairs of functions are inverses of each other.
$f(x)=0.75 x^{2}+2 ; g(x)=\sqrt{\frac{4(x-2)}{3}}$

AG
Ankit Gupta
Numerade Educator
03:19

Problem 93

Determine whether or not the given pairs of functions are inverses of each other.
$f(x)=1.4 x^{3}+3.2 ; g(x)=\sqrt[3]{\frac{x-3.2}{1.4}}$

AG
Ankit Gupta
Numerade Educator
03:05

Problem 94

Determine whether or not the given pairs of functions are inverses of each other.
$f(x)=\sqrt{2.5 x+9.25}$
$g(x)=0.4 x^{2}-3.7, x \geq 0$

AG
Ankit Gupta
Numerade Educator
01:59

Problem 95

Determine whether or not the given pairs of functions are inverses of each other.
$f(x)=0.8 x^{1 / 2}+5.23$
$g(x)=1.25\left(x^{2}-5.23\right), x \geq 0$

AG
Ankit Gupta
Numerade Educator
02:45

Problem 96

Determine whether or not the given pairs of functions are inverses of each other.
$f(x)=2.5\left(x^{3}-7.1\right)$
$g(x)=\sqrt[3]{0.4 x+7.1}$

AG
Ankit Gupta
Numerade Educator
05:04

Problem 97

Match each function in Column A with its inverse from Column B.
Column $A$
(1) $y=5 x^{3}+10$
(2) $y=(5 x+10)^{3}$
(3) $y=5(x+10)^{3}$
(4) $y=(5 x)^{3}+10$
Column $B$
A. $y=\frac{\sqrt[3]{x}-10}{5}$
B. $y=\sqrt[3]{\frac{x}{5}}-10$
C. $y=\sqrt[3]{\frac{x-10}{5}}$
D. $y=\frac{\sqrt[3]{x-10}}{5}$

Caleb Wood
Caleb Wood
Numerade Educator
01:51

Problem 98

Examine the following table. Is it possible that $f$ and $g$ are inverses of each other? Why or why not?
$\begin{array}{|r|r|r|}\hline x & {f(x)} & {g(x)} \\ \hline 6 & {6} & {6} \\ {7} & {6.5} & {8} \\ {8} & {7} & {10} \\ {9} & {7.5} & {12} \\ {10} & {8} & {14} \\ {11} & {8.5} & {16} \\ {12} & {9} & {18} \\ \hline\end{array}$

AG
Ankit Gupta
Numerade Educator
05:25

Problem 99

Assume in Exercise 98 that $f$ and $g$ are both linear functions. Find equations for $f(x)$ and $g(x) .$ Are $f$ and $g$ inverses of each other?

AG
Ankit Gupta
Numerade Educator
02:32

Problem 100

Let $c(w)$ represent the cost of mailing a package that weighs $w$ pounds. Let $f(n)$ represent the weight, in pounds, of $n$ copies of a certain book. Explain what $(c \circ f)(n)$ represents.

Caleb Wood
Caleb Wood
Numerade Educator
03:52

Problem 101

Let $g(a)$ represent the number of gallons of sealant needed to seal a bamboo floor with area $a$. Let $c(s)$ represent the cost of $s$ gallons of sealant. Which composition makes sense: $(c \circ g)(a)$ or $(g \circ c)(s) ?$ What does it represent?

Caleb Wood
Caleb Wood
Numerade Educator