Stewart J. College Algebra: Concepts and Contexts
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384 CHAPTER 4
■
Logarithmic Functions and Exponential Models
example
8
A Function That Is Not One-to-One
Is the function one-to-one?
Solution
The function f is not one-to-one because, for example,
So the inputs 2 and have the same output 4.
■ NOW TRY EXERCISE 53 ■
The graph of a function can help us to easily decide whether it is one-to-one. If
a horizontal line intersects the graph of f at more than one point, then we see from
Figure 6 that there are inputs and whose corresponding outputs and
are the same (because they are the same height above the x-axis). This means that f
is not one-to-one.
These observations lead us to the following graphical test for deciding whether
a function is one-to-one.
Horizontal Line Test
f 1x
2
2f 1x
1
2x
2
x
1
- 2
f 122= 4andf 1- 22= 4
f 1x2= x
2
x⁄
y=Ï
0
x¤
f(x⁄) f(x¤)
x
y
figure 6 Horizontal Line Test
A function is one-to-one if and only if no horizontal line intersects its graph
more than once.
We use the Horizontal Line Test in the following examples.
example
9
Using the Horizontal Line Test
Determine whether the function is one-to-one.
(a) (b) (c)
Solution
We first sketch graphs of each of these functions as in Figure 7. Notice the difference
between the graph of f (whose domain is all real numbers) and the graph of (whose
domain is the nonnegative real numbers).
g
h1x 2= x
3
g1x 2= x
2
for x Ú 0f 1x 2= x
2
x
y
y y
x x
1
0
1
1
(a) f(x)=x™
Not one-to-one
(b) g(x)=x™ (x ⱖ 0)
One-to-one
(c) h(x)=x£
One-to-one
0
1
1
0
1
figure 7
SECTION 4.6
■
Working with Functions: Composition and Inverse 385
(a) From Figure 7(a) we see that there are horizontal lines that intersect the graph
of more than once. So by the Horizontal Line Test, f is not one-to-
one.
(b) From Figure 7(b) we see that no horizontal line intersects the graph of
more than once. So by the Horizontal Line Test, is
one-to-one.
(c) From Figure 7(c) we see that no horizontal line intersects the graph of
more than once. So by the Horizontal Line Test, h is one-to-one.
■ NOW TRY EXERCISE 71 ■
The function of Example 9 is not one-to-one, so there is no inverse
function for f. You can check that the inverse function for is the
function . Also, the inverse function of is the func-h1x 2= x
3
1x Ú 0 2g
-1
1x 2= 1x
1x Ú 0 2g1x 2= x
2
f 1x2= x
2
h1x 2= x
3
g1x Ú 0 2g1x2= x
2
f 1x2= x
2
2
■ Exponential and Logarithmic Functions as Inverse Functions
0
f(x)=a˛,
a>1
x
y
figure 8 is one-to one.f 1x2= a
x
Every exponential function , with , is a one-to-one function
by the Horizontal Line Test (see Figure 8 for the case in which ). So every ex-
ponential function has an inverse function. We can check that the logarithm function
is the inverse of f by the Inverse Function Property (as in Example 7):
Definition of f
Definition of g
Property of logarithms
Also,
Definition of g
Definition of f
Property of logarithms
So is the inverse of f by the Inverse Function Property.
Inverse Functions of Exponential Functions
g
= x
= a
log
a
x
f 1g1x 22= f 1log
a
x2
= x
= log
a
a
x
g1f 1x 22= g1a
x
2
g1x 2= log
a
x
a 7 1
a 7 0, a ⫽ 1f 1x 2= a
x
The exponential function and the logarithmic function
are inverse functions of each other. So
log
a
a
x
= xanda
log
a
x
= x
g1x 2= log
a
x
f 1x2= a
x
example
10
Checking That f and Are Inverses of Each Otherg
Show that and are inverses of each other.g1x 2= 10
x>2
f 1x2= log x
2
tion .h
-1
1x 2= 2
3
x
386 CHAPTER 4
■
Logarithmic Functions and Exponential Models
Solution
We need to check that and . We have
Definition of g
Definition of f
Rules of Exponents
log x and are inverse functions
Also
Definition of f
Definition of g
Laws of Logarithms
Simplify
log x and are inverse functions
This shows that f and are inverses of each other.
■ NOW TRY EXERCISE 41 ■
You may have noticed that the graphs of the exponential and logarithm functions
have similar shapes. In fact, the graph of is the mirror image of the
graph of , where the “mirror” is the line (Figure 9).
In general, the graph of is the reflection of the graph of f through the line
. This is because if , then , so if (s, t) is on the graph of f, then
(t, s) is on the graph of .f
-1
f
-1
1t 2= sf 1s2= ty = x
f
-1
y = xf 1x2= a
x
g1x 2= log
a
x
g
10
x
= x
= 10
log x
= 10
12log x2>2
= 10
1log x
2
2>2
g1f 1x 22= gAlog x
2
B
10
x
= x
= log 10
x
= logA10
x>2
B
2
f 1g1x 22= f A10
x>2
B
g1f 1x22= xf 1g1x22= x
x
y
y=a˛, a>1
y=log
a
x
y
=x
1
1
figure 9 Graphs of
and g1x 2= log
a
x
f 1x 2= a
x
4.6 Exercises
CONCEPTS
Fundamentals
1. If and , then _______.
2. If the rule of the function f is “Add 1” and the rule of the function is “Multiply by 2,”
then the rule of the function is “
_____________,” and the rule of the
function is “
______________.” Now express these functions
algebraically:
_______ _______
_______ _______
3. A function f is one-to-one if different inputs produce _______ outputs. You can tell
from the graph that a function is one-to-one by using the
_______ _______ Test.
4. (a) For a function to have an inverse, it must be
_______. So which one of the
following functions has an inverse?
(b) What is the inverse of the function that you chose in part (a)?
f 1x 2= x
2
,g1x 2= x + 2
N1x2= M1x2=
g1x2= f 1x2=
N1x 2= g1f 1x 22
M1x 2= f 1g1x 22
g
f 1g1222=f 15 2= 12g12 2= 5
x
g1x 2
1 3
2 5
3 4
4 1
5 3
SECTION 4.6
■
Working with Functions: Composition and Inverse 387
5. If the rule for the function f is “Multiply by 3, add 5, and then take the third power,” then
the rule for is “
______________.” We can express these functions algebraically as:
_______ _______
6. The inverse of is the function _______.
Think About It
7. If f and are linear functions, what kind of function is f composed with ? What kind of
function is ?
8. True or false?
(a) If f has an inverse, then is the same as .
(b) If f has an inverse, then .
9–12
■ The graphs of two functions f and are given. Use the graphs to evaluate the
indicated expressions.
g
f
-1
1f 1x 22= x
1
f 1x 2
f
-1
1x 2
f
-1
gg
f
-1
1x 2=f 1x 2= e
x
f
-1
1x 2=f 1x 2=
f
-1
x
0
4
4
f
g
y
x
0
4
4
y
SKILLS
x
f 1x 2
1 5
2 1
3 2
4 2
5 4
9. (a) (b)
10. (a) (b)
11. (a) (b)
12. (a) (b)
13–14
■ Two functions f and are given by tables. Complete the tables for the functions
“f composed with ” and “ composed with f.”
13.
gg
g
g1f 1g18 222f 1g1f 18222
g1g1622f 1f 1622
g1f 1022f 1g1022
g1f 1422f 1g1422
x
f 1g1x22
1
2
2
3
4
5
x
g1f 1x22
1
3
2
3
4
5
x
g1x 2
- 3
3
- 2
3
- 1
1
0 0
1
- 1
2
- 3
3
- 3
x
f 1x 2
- 3
0
- 2
1
- 1 - 1
0 2
1
- 2
2 3
3
- 3
388 CHAPTER 4
■
Logarithmic Functions and Exponential Models
14.
15–22 ■ Two functions f and are given.
(a) Give a verbal description of the functions and .
(b) Find algebraic expressions for the functions and .
(c) Evaluate and .
15.
16.
17.
18.
19.
20.
21.
22.
23–28
■ A verbal description of the rule of a function f is given.
(a) Give a verbal description of the inverse function .
(b) Find algebraic expressions for and .
(c) Verify that and .
23. Multiply the input by 3, then subtract 6.
24. Multiply the input by 5, then add 2.
25. Add 4 to the input, then multiply by .
26. Subtract 2 from the input, then divide by 4.
27. Square the input, then multiply by and add 3.
28. Square the input, add 1, then divide by 7.
29–32
■ Assume that f is a one-to-one function.
29. If , find .
30. If , find .
31. If , find .
32. If , find .f
122f
-1
14 2= 2
f
1- 12f
-1
13 2=-1
f
-1
118 2f 1- 52= 18
f
-1
17 2f 122= 7
- 2
1
2
f
-1
1f 1x 22= xf 1f
-1
1x 22= x
f
-1
1x 2f 1x2
f
-1
f 1x 2= log x,g1x2= x
4
+ 1
f 1x 2= 3x + 5,
g1x 2= ln x
f 1x 2= 2x - 4,
g1x 2= 10
x
f 1x 2= e
x
,g1x 2= x + 3
f 1x 2= x - 2,
g1x 2= x
2
f 1x 2= 2x
2
,g1x 2= x - 1
f 1x 2= 5x,
g1x 2= x - 1
f 1x 2= x + 2,
g1x 2= 3x
N1- 22M132
N1x 2M1x2
N1x 2= g1f 1x 22M1x 2= f 1g1x22
g
x
f 1g1x22
- 3 - 3
- 2
- 1
0
1
2
3
x
g1f 1x22
- 3
0
- 2
- 1
0
1
2
3
x 01234 5
f 1x 2
1.1 2.4 5.8 3.6 2.1 2.4
x 0123 4 5
f 1x 2
1
1
3
3927 81
x
f 1x 2
0
1
2
3
4
5
6
7
⫺3
5
12
⫺1
10
6
SECTION 4.6
■
Working with Functions: Composition and Inverse 389
33–34 ■ A table of values for a one-to-one function f is given. Find
(a) (b) (c) (d)
33. 34.
f
-1
110 2f
-1
1f 1422f
-1
15 2f 132
37–44
■ Show that f and are inverses of each other (as in Example 6).
37.
38.
39.
40.
41. 42.
43. 44.
45–56
■ A function is given by a table of values, a graph, a formula, or a verbal
description. Determine whether the function is one-to-one.
45.
46.
f 1x 2= ln x + 2;
g1x 2= e
x - 2
f 1x 2= log
5
x
2
;g1x 2= 5
x>2
f 1x 2= 10
x>3
;g1x 2= log x
3
f 1x 2= 10
#
4
x
;g1x 2= log
4
a
x
10
b
f 1x 2=
1
x - 1
;g1x 2=
1
x
+ 1
f 1x 2= x
3
+ 1;g1x2= 2
3
x - 1
f 1x 2=
3 - x
4
;g1x 2= 3 - 4x
f 1x 2= 2x - 5;
g1x 2=
x + 5
2
g
x
f 1x 2
⫺1
0
1
2
3
4
5
3
5
⫺2
0
10
1
6
35–36 ■ The graph of a function is given. Use the graph to find the indicated values.
35. (a) , (b) , (c) 36. (a) , (b) , (c) g
-1
16 2g
-1
15 2g
-1
12 2f
-1
16 2f
-1
14 2f
-1
12 2
x
0
4
4
f
y
g
x
0
4
4
y
390 CHAPTER 4
■
Logarithmic Functions and Exponential Models
47. 48.
0
x
y
0
x
y
0
x
y
0
x
y
49. 50.
51. 52.
53. 54.
55.
h(t) is the height of a basketball t seconds after it is thrown in a free-throw shot.
56. S(x) is your shoe size at age x.
57–70
■ Find the inverse function of f.
57. 58.
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70.
71–76
■ Draw the graph of f and use it to determine whether the function is one-to-one.
71. 72.
73. 74.
75. 76.
77. Multiple Discounts You have a $50 coupon from the manufacturer good for the
purchase of a cell phone. The store where you are purchasing your cell phone is offering
a 20% discount on all cell phones. Let x represent the sticker price of the cell phone.
(a) Suppose that only the 20% discount applies. Find a function f that models the
purchase price of the cell phone as a function of the sticker price x.
(b) Suppose that only the $50 coupon applies. Find a function that models the
purchase price of the cell phone as a function of the sticker price x.
g
f 1x 2= x
#
0 x 0f 1x2= 0 x 0- 0 x - 6 0
f 1x 2= 2x
3
- 4x + 1f 1x 2=
x + 12
x - 6
f 1x 2= x
3
+ xf 1x2= x
3
- x
f 1x 2= 2
x
3
f 1x 2= ln1x - 32
f 1x 2= log
3
12x 2f 1x 2= e
0.5x
f 1x 2= 10
3x
f 1x 2= log
2
1x + 12
f 1x 2=
1 + x
3 - x
f 1x 2=
x - 2
x + 2
f 1x 2= 2
3
x + 2f 1x 2= x
3
- 4
f 1x 2=
1
x
f 1x 2=
x
2
f 1x 2= 3 - 5xf 1x 2= 4x + 7
g1x 2= x
4
+ 5g1x 2= x
2
- 2x
f 1x 2= 3x - 2f 1x 2=-2x + 4
CONTEXTS
(c) If you can use the coupon and the discount, then the purchase price is either
or , depending on the order in which they are applied to the sticker price.
Find both and . Which composition gives the lower price?
78. Multiple Discounts An appliance dealer advertises a 10% discount on all his
washing machines. In addition, the manufacturer offers a $100 rebate on the purchase of
a washing machine. Let x represent the sticker price of the washing machine.
(a) Suppose that only the 10% discount applies. Find a function f that models the
purchase price of the washer as a function of the sticker price x.
(b) Suppose that only the $100 rebate applies. Find a function that models the
purchase price of the washer as a function of the sticker price x.
(c) Find and . What do these functions represent? Which is the better
deal?
79. Carbon Dioxide Levels Many scientists believe that the increase of carbon dioxide
in the atmosphere is a major contributor to global warming. The EPA estimates
that 1 gallon of gas produces on average about 19 pounds of . Raul owns a car that
gets 30 miles per gallon. Raul is concerned about how his driving habits affect the
environment, so he decides to calculate his car’s emissions for several different
mileages.
(a) Find a rule for the function that gives the number of gallons that Raul uses in
driving x miles.
(b) Find a rule for the function f that gives the weight of emissions when x
gallons of gasoline are used.
(c) Evaluate the composite function . What does this represent?
(d) Use your answers to parts (a)–(c) to complete the table.
f 1g1x22
f 1x 2CO
2
g1x 2g
CO
2
CO
2
1CO
2
2
g1f 1x22f 1g1x 22
g
g1f 1x22f 1g1x 22
g1f 1x22
f 1g1x22
Miles driven
x
Gallons
g1x 2 f 1g1x22
12,000
400 7600
24,000
36,000
50,000
100,000
SECTION 4.6
■
Working with Functions: Composition and Inverse 391
80. Biodiesel Fuels Scientists have begun exploring the use of biodiesel fuel (plant-based
diesel fuel) as a substitute for fossil fuel. One possible source for biodiesel is oil
extracted from the seeds of the species Jatropha curcsa, a drought-resistant bushy tree
that is indigenous to India. In a study assessing the viability of using these trees as a
biofuel, scientists gathered data on the number of seeds per tree at various elevations
and the oil content of those seeds. Some of their data are shown in the tables on the
following page. The first table gives the average seed yield per tree at each elevation,
and the second gives the oil content of a seed produced by a tree bearing the given
number of seeds. Note that as elevation increases, so does seed production per tree, but
the more seeds a tree produces, the lower the oil content of those seeds.
Let be the average number of seeds per tree at elevation x meters, and let
be the amount of oil in each seed from a tree producing n seeds.
(a) Let . What does the function D represent? Complete the third table,
giving the values of .D1x 2
D1x 2= C1S1x 22
C1n 2
S1x 2
KAMBOU SIA/AFP/Getty Images
392 CHAPTER 4
■
Logarithmic Functions and Exponential Models
(b) Explain why the product gives the amount of oil produced per tree at
elevation x.
D1x 2
#
S1x 2
Elevation (m)
x
Seeds/tree
S1x 2
400 330
600 360
750 410
900 600
1000 670
Seeds/tree
n
Oil/seed (dL)
C1n 2
330 4.3
360 4.1
410 3.8
600 3.3
670 3.1
Elevation (m)
x
Oil/seed (dL)
D1x 2= C1S1x 22
400
4.3
600
750
900
1000
81. Multiple Discounts A car dealership advertises a 15% discount on all its new cars. In
addition, the manufacturer offers a $1000 rebate on the purchase of a new car. Let x
represent the sticker price of the car.
(a) Suppose that only the 15% discount applies. Find a function f that models the
purchase price of the car as a function of the sticker price x.
(b) Suppose that only the $1000 rebate applies. Find a function that models the
purchase price of the car as a function of the sticker price x.
(c) Find a formula for .
(d) Find . What does represent?
(e) Find . What does your answer represent?
82. Health Care Spending The function is an exponential model
for annual U.S. health care expenditures per capita, where x is the number of years since
1990 (see Exercise 39 in Section 3.1).
(a) Find . What does represent?
(b) Use the inverse function to predict when annual U.S. health care expenditures
per capita reaches $7000.
(c) Search the Internet to find when the reported annual health care expenditures
reached $7000. Compare with your results in part (b).
83. Population of India The function is an exponential model for
the population of India, where x is the number of years since 1990 and the population is
measured in millions (see Exercise 45 in Section 3.2).
(a) Find . What does represent?
(b) Use the inverse function to predict when the population of India reaches
1.1 billion.
(c) Search the Internet to find when the population of India actually did reach 1.1 billion.
Compare with your results in part (b).
84. Service Fee For his services, a private investigator charges a $500 retention fee plus
$80 per hour. Let x represent the number of hours the investigator spends on a case.
(a) Find a function f that models the investigator’s fee as a function of x.
(b) Find . What does represent?
(c) Find . What does your answer represent?
85. Torricelli’s Law A tank holds 100 gallons of water, which drains from a leak at the
bottom, causing the tank to empty in 40 minutes. Torricelli’s Law relates the volume
of water remaining in the tank after x minutes by the function
F 1x2= 100 a1 -
x
40
b
2
F 1x 2
f
-1
11220 2
f
-1
f
-1
f
-1
f
-1
f
-1
f 1x 2= 846.311.0312
x
f
-1
f
-1
f
-1
f 1x 2= 281311.05782
x
H
-1
113,000 2
H
-1
H
-1
H1x 2= f 1g1x 22
g
R
R
R
R
CHAPTER 4
■
Review 393
(a) Find . What physical quantities (volume or time) do the inputs and outputs of
represent?
(b) Find . What does your answer represent?F
-1
115 2
F
-1
F
-1
86. Blood Flow As blood moves through a vein or artery, its velocity is greatest along
the central axis and decreases as the distance r from the central axis increases (see the
figure below). For an artery with radius 0.5 cm, (in cm/s) is given as a function of r by
(a) Find . What does represent?
(b) Find . What does your answer represent?√
-1
10.9 2
√
-1
√
-1
√1r2= 4.810.25 - r
2
2
√
√
r
0.5 cm
CONCEPT CHECK
Make sure you understand each of the ideas and concepts that you learned in this chapter,
as detailed below section by section. If you need to review any of these ideas, reread the
appropriate section, paying special attention to the examples.
4.1 Logarithmic Functions
The logarithm base a of a number is the exponent to which we must raise the base
a to get the number. In symbols,
When the base is 10, we usually don’t write the a:
Logarithms have the following basic properties:
1. 2. 3. 4.
The logarithmic function with base a is the function
f 1x2= log
a
xa 7 0,a ⫽ 1
a
log
a
x
= xlog
a
a
x
= xlog
a
a = 1log
a
1 = 0
log x = ymeans10
y
= x
log
a
x = ymeansa
y
= x
CHAPTER 4 REVIEW
CHAPTER 4