E. Applications of Inverse Functions
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CHAPTER 5 Exponential and Logarithmic Functions
5.1 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
ᮣ
1. A function is one-to-one if each
coordinate
corresponds to exactly
first coordinate.
2. If every
line intersects the graph of a
function in at most
point, the function is
one-to-one.
3. A certain function is defined by the ordered pairs
1Ϫ2, Ϫ112, 10, Ϫ52, 12, 12 , and (4, 19). The inverse
function is
.
4. To find f Ϫ1 using the algebraic method, we (1) use
instead of f(x), (2)
x and y,
(3)
for y and replace y with f Ϫ1 1x2 .
5. State true or false and explain why: To show that g
is the inverse function for f, simply show that
1 f ؠg21x2 ϭ x. Include an example in your
response.
6. Discuss/Explain why no inverse function exists for
f 1x2 ϭ 1x ϩ 32 2 and g1x2 ϭ 24 Ϫ x2. How would
the domain of each function have to be restricted to
allow for an inverse function?
DEVELOPING YOUR SKILLS
Determine whether each graph given is the graph of a
one-to-one function. If not, give examples of how the
definition of one-to-oneness is violated.
7.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
9.
5
4
3
2
1
11.
y
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
10.
1 2 3 4 5 x
12.
1 2 3 4 5 x
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
15. 51Ϫ7, 42, 1Ϫ1, 92, 10, 52, 1Ϫ2, 12, 15, Ϫ526
16. 519, 12, 1Ϫ2, 72, 17, 42, 13, 92, 12, 726
17. 51Ϫ6, 12, 14, Ϫ92, 10, 112, 1Ϫ2, 72, 1Ϫ4, 52, 18, 126
1 2 3 4 5 x
y
5
4
3
2
1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
14.
Determine whether the functions given are one-to-one.
If not, state why.
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
y
5
4
3
2
1
1 2 3 4 5 x
y
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
5
4
3
2
1
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
y
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
8.
13.
1 2 3 4 5 x
18. 51Ϫ6, 22, 1Ϫ3, 72, 18, 02, 112, Ϫ12, 12, Ϫ32, 11, 326
Determine if the functions given are one-to-one by noting
the function family to which each belongs and mentally
picturing the shape of the graph. If a function is not one-toone, discuss how the definition of one-to-oneness is violated.
19. f 1x2 ϭ 3x Ϫ 5
20. g1x2 ϭ 1x ϩ 22 3 Ϫ 1
23. s1t2 ϭ 12t Ϫ 1 ϩ 5
3
24. r1t2 ϭ 2
tϩ1Ϫ2
25. y ϭ 3
26. y ϭ Ϫ2x
21. h1x2 ϭ ϪͿx Ϫ 4Ϳ ϩ 3
22. p1t2 ϭ 3t2 ϩ 5
For Exercises 27 to 30, find the inverse function of the
one-to-one functions given.
27. f 1x2 ϭ 51Ϫ2, 12, 1Ϫ1, 42, 10, 52, 12, 92, 15, 1526
28. g1x2 ϭ 51Ϫ2, 302, 1Ϫ1, 112, 10, 42, 11, 32, 12, 226
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Section 5.1 One-to-One and Inverse Functions
29. v(x) is defined by the ordered pairs shown.
For each function f(x) given, prove (using a composition)
that g(x) ϭ f Ϫ1(x).
53. f 1x2 ϭ Ϫ2x ϩ 5, g1x2 ϭ
54. f 1x2 ϭ 3x Ϫ 4, g1x2 ϭ
30. w(x) is defined by the ordered pairs shown.
xϪ5
Ϫ2
xϩ4
3
3
55. f 1x2 ϭ 2
x ϩ 5, g1x2 ϭ x3 Ϫ 5
3
56. f 1x2 ϭ 2
x Ϫ 4, g1x2 ϭ x3 ϩ 4
57. f 1x2 ϭ 23x Ϫ 6, g1x2 ϭ 32x ϩ 9
58. f 1x2 ϭ 45x ϩ 6, g1x2 ϭ 54x Ϫ
15
2
59. f 1x2 ϭ x2 Ϫ 3; x Ն 0, g1x2 ϭ 1x ϩ 3
60. f 1x2 ϭ x2 ϩ 8; x Ն 0, g1x2 ϭ 1x Ϫ 8
Find the inverse function using diagrams similar to
those illustrated in Example 2. Check the result using
three test points.
31. f 1x2 ϭ x ϩ 5
32. g1x2 ϭ x Ϫ 4
4
33. p1x2 ϭ Ϫ x
5
34. r1x2 ϭ
35. f 1x2 ϭ 4x ϩ 3
3x
4
36. g1x2 ϭ 5x Ϫ 2
3
37. t1x2 ϭ 2
xϪ4
3
38. s1x2 ϭ 2
xϩ2
State the domain and range of f (x), then use the
algebraic method to find the inverse function and state
its domain and range. Finally, find any three ordered
pairs (a, b) that satisfy f, and verify the ordered pairs
(b, a) satisfy the equation for f ؊1.
3
39. f 1x2 ϭ 2
xϪ2
3
40. f 1x2 ϭ 2
xϩ3
8
43. f 1x2 ϭ
xϩ2
12
44. f 1x2 ϭ
xϪ1
x
45. f 1x2 ϭ
xϩ1
xϩ2
46. f 1x2 ϭ
1Ϫx
41. f 1x2 ϭ x3 ϩ 1
42. f 1x2 ϭ x3 Ϫ 2
The functions given in Exercises 47 through 52 are not
one-to-one. (a) Determine a domain restriction that
preserves all range values and creates a one-to-one
function, then state the new domain and range. (b) State
the domain and range of the inverse function and find
its equation.
47. f 1x2 ϭ 1x ϩ 52
49. v1x2 ϭ
2
8
1x Ϫ 32 2
51. p1x2 ϭ 1x ϩ 42 2 Ϫ 2
48. g1x2 ϭ x ϩ 3
2
50. V1x2 ϭ
52. q1x2 ϭ
4
ϩ2
x2
4
ϩ1
1x Ϫ 22 2
Find the inverse of each function f(x) given, then prove
(by composition) your inverse function is correct. Note the
domain and range of f in each case is all real numbers.
61. f 1x2 ϭ 3x Ϫ 5
62. f 1x2 ϭ 5x ϩ 4
63. f 1x2 ϭ
64. f 1x2 ϭ
xϪ5
2
xϩ4
3
65. f 1x2 ϭ 12x Ϫ 3
66. f 1x2 ϭ 23x ϩ 1
3
69. f 1x2 ϭ 2
2x ϩ 1
3
70. f 1x2 ϭ 2
3x Ϫ 2
67. f 1x2 ϭ x3 ϩ 3
71. f 1x2 ϭ
1x Ϫ 12 3
8
68. f 1x2 ϭ x3 Ϫ 4
72. f 1x2 ϭ
1x ϩ 32 3
Ϫ27
The functions given in Exercises 73 through 78 are oneto-one. State the implied domain of each function given,
and use these to state the domain and range of the
inverse function. Then find the inverse and prove by
composition that your inverse is correct.
73. f 1x2 ϭ 13x ϩ 2
74. g1x2 ϭ 12x Ϫ 5
75. p1x2 ϭ 2 1x Ϫ 3
76. q1x2 ϭ 41x ϩ 1
77. v1x2 ϭ x ϩ 3; x Ն 0
78. w1x2 ϭ x2 Ϫ 1; x Ն 0
2
Determine the domain and range for each one-to-one
function whose graph is given, and use this information
to state the domain and range of the inverse function.
Then sketch in the line y ؍x, estimate the location of
two or more points on the graph, and use this
information to graph f ؊1(x) on the same grid.
79.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
80.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
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81.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
82.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
83.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
1 2 3 4 5 x
Ϫ2
Ϫ3
Ϫ4
Ϫ5
84.
y
5
4
3
2
1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
Ϫ4
Ϫ5
1 2 3 4 5 x
For the functions given, (a) find f ؊1(x), then use your calculator to verify they are inverses by (b) using ordered pairs,
(c) composing the functions, and (d) showing their graphs are symmetric to y ؍x.
85. f 1x2 ϭ 2x ϩ 1
87. h1x2 ϭ
ᮣ
x
xϩ1
88. j1x2 ϭ 2 2x ϩ 9 Ϫ 6
WORKING WITH FORMULAS
89. The height of a projected image: f (x) ؍12x ؊ 8.5
The height of an image projected on a screen
is given by the formula shown, where f 1x2
represents the actual height of the image on the
projector (in centimeters) and x is the distance of
the projector from the screen (in centimeters).
(a) When the projector is 80 cm from the screen,
how large is the image? (b) Show that the inverse
function is f Ϫ1 1x2 ϭ 2x ϩ 17, then input your
answer from part (a) and comment on the result.
What information does the inverse function
give?
ᮣ
86. g1x2 ϭ x2 ϩ 1; x Ն 0
90. The radius of a sphere: r(x) ؍
3x
B4
3
In generic form, the radius of a sphere is given by
the formula shown, where r(x) represents the radius
and x represents the volume of the sphere in cubic
units. (a) If a weather balloon that is roughly
spherical holds 14,130 in3 of helium, what is the
radius of the balloon (use Ϸ 3.142 ? (b) Show that
the inverse function is rϪ1 1x2 ϭ 43x3, then input
your answer from part (a) and comment on the
result. What information does the inverse function
give?
APPLICATIONS
91. Temperature and altitude: The temperature (in degrees Fahrenheit) at a given altitude can be approximated by
the function f 1x2 ϭ Ϫ72x ϩ 59, where f(x) represents the temperature and x represents the altitude in thousands of
feet. (a) What is the approximate temperature at an altitude of 35,000 ft (normal cruising altitude for commercial
airliners)? (b) Find f Ϫ1 1x2, and state what the independent and dependent variables represent. (c) If the
temperature outside a weather balloon is Ϫ18°F, what is the approximate altitude of the balloon?
92. Fines for speeding: In some localities, there is a set formula to determine the amount of a fine for exceeding
posted speed limits. Suppose the amount of the fine for exceeding a 50 mph speed limit was given by the
function f 1x2 ϭ 12x Ϫ 560 (x 7 50) where f (x) represents the fine in dollars for a speed of x mph. (a) What is
the fine for traveling 65 mph through this speed zone? (b) Find f Ϫ1 1x2, and state what the independent and
dependent variables represent. (c) If a fine of $172 were assessed, how fast was the driver going through this
speed zone?
93. Effect of gravity: Due to the effect of gravity, the distance an object has fallen after being dropped is given by
the function f 1x2 ϭ 16x2; x Ն 0, where f (x) represents the distance in feet after x sec. (a) How far has the object
fallen 3 sec after it has been dropped? (b) Find f Ϫ1 1x2, and state what the independent and dependent variables
represent. (c) If the object is dropped from a height of 784 ft, how many seconds until it hits the ground (stops
falling)?
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94. Area and radius: In generic form, the area of a
circle is given by f 1x2 ϭ x2, where f (x) represents
the area in square units for a circle with radius x.
(a) A pet dog is tethered to a stake in the backyard.
If the tether is 10 ft long, how much area does the
dog have to roam (use Ϸ 3.14)? (b) Find f Ϫ1 1x2,
and state what the independent and dependent
variables represent. (c) If the owners want to allow
the dog 1256 ft2 of area to live and roam, how long
a tether should be used?
95. Volume of a cone: In
r
generic form, the volume
of an equipoise cone
h
(height equal to radius) is
hϭr
1
3
given by f 1x2 ϭ 3x ,
where f(x) represents
the volume in units3 and
x represents the height of the cone. (a) Find the
volume of such a cone if r ϭ 30 ft (use Ϸ 3.142.
(b) Find f Ϫ1 1x2, and state what the independent and
ᮣ
491
dependent variables represent. (c) If the volume of
water in the cone is 763.02 ft3, how deep is the
water at its deepest point?
96. Wind power: The power delivered by a certain
wind-powered generator can be modeled by the
x3
, where f (x) is the horsepower
function f 1x2 ϭ
2500
(hp) delivered by the generator and x represents
the speed of the wind in miles per hour. (a) Use
the model to determine how much horsepower is
generated by a 30 mph wind. (b) The person
monitoring the output of the generators (wind
generators are usually erected in large numbers)
would like a function that gives the wind speed
based on the horsepower readings on the gauges
in the monitoring station. For this purpose, find
f Ϫ1 1x2 and state what the independent and dependent
variables represent. (c) If gauges show 25.6 hp is
being generated, how fast is the wind blowing?
EXTENDING THE CONCEPT
97. For a deeper understanding of the algebraic method
for finding an inverse, suppose a function f is
defined as f 1x2:5 1x, y2 | y ϭ 3x Ϫ 66. We can then
define the inverse as f Ϫ1: 5 1x, y2 | x ϭ 3y Ϫ 66,
having interchanged x and y in the equation
portion. The equation for f Ϫ1 is not in standard
form, but (x, y) still represents all ordered pairs
satisfying either equation. Solving for y gives
x
f Ϫ1: e 1x, y2 ` y ϭ ϩ 2 f , and demonstrates the
3
role of steps 2, 3, and 4 of the method. (a) Find five
ordered pairs that satisfy the equation for f, then
(b) interchange their coordinates and show they
satisfy the equation for f Ϫ1.
ᮣ
Section 5.1 One-to-One and Inverse Functions
98. By inspection, which of the following is the inverse
2
1 5 4
function for f 1x2 ϭ ax Ϫ b ϩ ?
3
2
5
a. f Ϫ1 1x2 ϭ
2
4
5 1
ax Ϫ b Ϫ
3
5
B2
3 5
5
b. f Ϫ1 1x2 ϭ 2 1x Ϫ 22 Ϫ
2
4
3 5
1
5
ax ϩ b Ϫ
c. f Ϫ1 1x2 ϭ
2B
2
4
4
1
5 3
ax Ϫ b ϩ
d. f Ϫ1 1x2 ϭ
B2
5
2
MAINTAINING YOUR SKILLS
99. (R.3) Write as many of the following formulas as you can from memory:
a. perimeter of a rectangle
d. volume of a cone
b. area of a circle
e. circumference of a circle
c. volume of a cylinder
f. area of a triangle
100. (3.2) Given f 1x2 ϭ x2 Ϫ x Ϫ 2, solve the
inequality f 1x2 Յ 0 using the x-intercepts and endbehavior of the graph.
101. (3.4) For the function y ϭ 2 1x ϩ 3, find the
average rate of change between x ϭ 1 and x ϭ 2,
and between x ϭ 4 and x ϭ 5. Which is greater?
Why?
g. area of a trapezoid
h. volume of a sphere
i. Pythagorean theorem
102. (R.4) Solve the following cubic equations by
factoring:
a. x3 Ϫ 5x ϭ 0
b. x3 Ϫ 7x2 Ϫ 4x ϩ 28 ϭ 0
c. x3 Ϫ 3x2 ϭ 0
d. x3 Ϫ 3x2 Ϫ 4x ϭ 0
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CHAPTER 5 Exponential and Logarithmic Functions
5.2
Exponential Functions
LEARNING OBJECTIVES
In Section 5.2 you will see
how we can:
A. Evaluate an exponential
Demographics is the statistical study of human populations. In this section, we introduce the family of exponential functions, which are widely used to model population
growth or decline with additional applications in science, engineering, and many other
fields. As with other functions, we begin with a study of the graph and its characteristics.
function
B. Graph general
exponential functions
C. Graph base-e
exponential functions
D. Solve exponential
equations and
applications
A. Evaluating Exponential Functions
In the boomtowns of the old west, it was not
uncommon for a town to double in size every year
(at least for a time) as the lure of gold drew more
and more people westward. When this type of
growth is modeled using mathematics, exponents
play a lead role. Suppose the town of Goldsboro
had 1000 residents when gold was first discovered.
After 1 yr the population doubled to 2000 residents. The next year it doubled again to 4000, then
again to 8000, then to 16,000 and so on. You probably recognize the digits in blue as
powers of two (indicating the population is doubling), with each one multiplied by 1000
(the initial population). This suggests we can model the relationship using
P1x2 ϭ 1000 # 2x
where P(x) is the population after x yr. Further, we can evaluate this function, called
an exponential function, for fractional parts of a year using rational exponents. The
population of Goldsboro one-and-a-half years after the gold rush was
3
3
P a b ϭ 1000 # 22
2
ϭ 1000 # 1 122 3
Ϸ 2828 people
In general, exponential functions are defined as follows.
WORTHY OF NOTE
To properly understand the
exponential function and its graph
requires that we evaluate f 1x2 ϭ 2x
even when x is irrational. For
example, what does 215 mean?
While the technical details require
calculus, it can be shown that
successive approximations of 215
as in 22.2360, 22.23606, 22.236067, . . .
approach a unique real number,
and that f 1x2 ϭ 2x exists for all real
numbers x.
Exponential Functions
For b 7 0, b
1, and all real numbers x,
f 1x2 ϭ bx
defines the base b exponential function.
Limiting b to positive values ensures that outputs will be real numbers, and the
restriction b 1 is needed since y ϭ 1x is a constant function (1 raised to any power
is still 1). Specifically note the domain of an exponential function is all real numbers,
and that all of the familiar properties of exponents still hold. A summary of these properties follows. For a complete review, see Section R.2.
Exponential Properties
For real numbers a, b, m, and n, with a, b 7 0,
bm # bn ϭ bmϩn
1ab2 n ϭ an # bn
492
bm
ϭ bmϪn
bn
1
bϪn ϭ n
b
1bm 2 n ϭ bmn
b Ϫn
a n
a b ϭa b
a
b
5–14
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Section 5.2 Exponential Functions
EXAMPLE 1
ᮣ
493
Evaluating Exponential Functions
Evaluate each exponential function for x ϭ 2, x ϭ Ϫ1, x ϭ 12, and x ϭ . Use a
calculator for x ϭ , rounding to five decimal places.
4 x
a. f 1x2 ϭ 4x
b. g1x2 ϭ a b
9
Solution
ᮣ
a. For
f 1x2 ϭ 4x,
f 122 ϭ 42 ϭ 16
f 1Ϫ12 ϭ 4Ϫ1 ϭ
1
4
4 x
b. For g1x2 ϭ a b ,
9
4 2 16
g122 ϭ a b ϭ
9
81
4 Ϫ1 9
g1Ϫ12 ϭ a b ϭ
9
4
1
1
1
f a b ϭ 42 ϭ 14 ϭ 2
2
A. You’ve just seen how we
can evaluate an exponential
function
4
2
1
4 2
ϭ
ga b ϭ a b ϭ
A9
2
9
3
4
g12 ϭ a b Ϸ 0.07827
9
f 12 ϭ 4 Ϸ 77.88023
Now try Exercises 7 through 10
ᮣ
B. Graphing Exponential Functions
To gain a better understanding of exponential functions, we’ll graph examples of
y ϭ bx and note some of the characteristic features. Since b 1, it seems reasonable
that we graph one exponential function where b 7 1 and one where 0 6 b 6 1.
EXAMPLE 2
ᮣ
Graphing Exponential Functions with b Ͼ 1
Graph y ϭ 2x using a table of values.
Solution
ᮣ
To get an idea of the graph’s shape we’ll use integer values from Ϫ3 to 3 in our
table, then draw the graph as a continuous curve, since the function is defined for
all real numbers.
y
x
y ؍2x
Ϫ3
2Ϫ3 ϭ 18
Ϫ2
2Ϫ2 ϭ 14
Ϫ1
2Ϫ1 ϭ 12
0
20 ϭ 1
1
2 ϭ2
2
22 ϭ 4
3
23 ϭ 8
(3, 8)
8
4
(2, 4)
(1, 2)
1
(0, 1)
Ϫ4
4
x
WORTHY OF NOTE
As in Example 2, functions that are
increasing for all x ʦ D are said to
be monotonically increasing or
simply monotonic functions. The
function in Example 3 is monotonically decreasing.
Now try Exercises 11 and 12
ᮣ
Several important observations can now be made. First note the x-axis (the line
y ϭ 0) is a horizontal asymptote for the function, because as x S Ϫq, y S 0. Second,
the function is increasing over its entire domain, giving the function a range of
y ʦ 10, q 2.
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CHAPTER 5 Exponential and Logarithmic Functions
EXAMPLE 3
ᮣ
Solution
ᮣ
Graphing Exponential Functions with 0 Ͻ b Ͻ 1
Graph y ϭ 1 12 2 x using a table of values.
Using properties of exponents, we can write 1 12 2 x as 1 21 2 Ϫx ϭ 2Ϫx. Again using
integers from Ϫ3 to 3, we plot the ordered pairs and draw a continuous curve.
y
x
y ؍2؊x
Ϫ3
2Ϫ1Ϫ32 ϭ 23 ϭ 8
Ϫ2
2Ϫ1Ϫ22 ϭ 22 ϭ 4
Ϫ1
2Ϫ1Ϫ12 ϭ 21 ϭ 2
(Ϫ3, 8)
(Ϫ2, 4)
20 ϭ 1
0
Ϫ1
ϭ
1
2
2
2Ϫ2 ϭ
3
2Ϫ3 ϭ
8
4
(Ϫ1, 2)
(0, 1)
1
2
1
4
1
8
Ϫ4
x
4
Now try Exercises 13 and 14
ᮣ
We note this graph is also asymptotic to the x-axis, but decreasing on its domain.
In addition, both y ϭ 2x and y ϭ 2Ϫx ϭ 1 12 2 x have a y-intercept of (0, 1) and both are
one-to-one, which suggests that an inverse function can be found. Finally, observe that
y ϭ bϪx is a reflection of y ϭ bx across the y-axis, a property that indicates these basic
graphs might also be transformed in other ways, as were the toolbox functions. The
characteristics of exponential functions are summarized here:
f 1x2 ؍bx, b Ͼ 0 and b
1
• one-to-one function
• y-intercept (0, 1)
• domain: x ʦ ޒ
• range: y ʦ 10, q 2
• increasing if b 7 1
• decreasing if 0 6 b 6 1
• asymptotic to the x-axis (the line y ϭ 0)
Figure 5.18
Figure 5.19
y
y
f(x) ϭ bx
0ϽbϽ1
f(x) ϭ
bϾ1
bx
WORTHY OF NOTE
When an exponential function is
increasing, it can be referred to as a
“growth function.” When decreasing,
it is often called a “decay function.”
Each of the graphs shown in
Figures 5.18 and 5.19 should now
be added to your repertoire of
basic functions, to be sketched
from memory and analyzed or used
as needed.
(1, b)
(0, 1)
(0, 1)
Ϫ4
4
x
Ϫ4
1, 1b
4
x
Just as the graph of a quadratic function maintains its parabolic shape regardless
of the transformations applied, exponential functions will also maintain their general shape and features. Any sum or difference applied to the basic function
1y ϭ bx Ϯ k vs. y ϭ bx 2 will cause a vertical shift in the same direction as the sign, and
any change to input values 1y ϭ bxϩh vs. y ϭ bx 2 will cause a horizontal shift in a
direction opposite the sign. For cases where multiple transformations are to be
applied, refer to the sequence outlined on page 209.
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Section 5.2 Exponential Functions
EXAMPLE 4
ᮣ
Graphing Exponential Functions Using Transformations
Graph F1x2 ϭ 2xϪ1 ϩ 2 using transformations of
the basic function f 1x2 ϭ 2x (not by simply plotting
points). Clearly state what transformations are
applied.
Solution
ᮣ
y
F(x) = 2x is shifted
1 unit right
2 units up
The graph of F is that of the basic function
f 1x2 ϭ 2x with a horizontal shift 1 unit right and
a vertical shift 2 units up. With this in mind the
horizontal asymptote also shifts from y ϭ 0 to
y ϭ 2 and (0, 1) shifts to (1, 3). The y-intercept of
F is at (0, 2.5):
(0, 2.5)
(3, 6)
(1, 3)
yϭ2
Ϫ4
4
x
F102 ϭ 2102Ϫ1 ϩ 2
ϭ 2Ϫ1 ϩ 2
1
ϭ ϩ2
2
ϭ 2.5
B. You’ve just seen how
we can graph general
exponential functions
To help sketch a more accurate graph, the point (3, 6) can be used: F132 ϭ 6.
Now try Exercises 15 through 30
ᮣ
C. The Base-e Exponential Function: f(x) ؍ex
In nature, exponential growth occurs when the rate of change in a population’s growth
is in constant proportion to its current size. Using the rate of change notation,
¢P
ϭ kP, where k is a constant. For the city of Goldsboro, we know the population at
¢t
time t is given by P1t2 ϭ 1000 # 2t, but have no information on this value of k (see
Exercise 90). We can actually rewrite this function, and other exponential functions,
using a base that gives the value of k directly and without having to apply the difference quotient. This new base is an irrational number, symbolized by the letter e. In
Section 5.6 we’ll develop the number e in the context of compound interest, while
making numerous references to our discussion here, where we define e as follows.
WORTHY OF NOTE
Just as the ratio of a circle’s
circumference to its diameter is an
irrational number symbolized by ,
the irrational number that results
1 x
from a1 ϩ b for infinitely large
x
x is symbolized by e. Writing
exponential functions in terms of e
simplifies many calculations in
advanced courses, and offers
additional advantages in applications
of exponential functions.
The Number e
For x 7 0,
1 x
as x S q, a1 ϩ b S e
x
1 x
In words, e is the number that a1 ϩ b approaches as x becomes infinitely large.
x
1 x
It has been proven that as x grows without bound, a1 ϩ b indeed approaches the
x
unique, irrational number that we have named e. Table 5.5 gives approximate values of
the expression for selected values of x, and shows e Ϸ 2.71828 to five decimal places.
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CHAPTER 5 Exponential and Logarithmic Functions
The result is the base-e exponential
function: f 1x2 ϭ ex, also called the natural
exponential function. Instead of having to
enter a decimal approximation when computing with e, most calculators have an “ex ” key,
usually as the 2nd function for the key
marked LN . To find the value of e2, use the
keystrokes 2nd LN 2 )
, and the calculator display should read 7.389056099. Note
the calculator supplies the left parenthesis for
the exponent, and you must supply the right.
See Figure 5.20.
Figure 5.20
Table 5.5
10
ᮣ
Solution
ᮣ
2.59
2.705
100
1000
ENTER
EXAMPLE 5
Approximate Value
11 ؉ 1x 2 x
x
2.7169
2.71815
10,000
100,000
2.718268
2.7182805
1,000,000
10,000,000
2.71828169
Evaluating the Natural Exponential Function
Use a calculator to evaluate f 1x2 ϭ ex for the values of x given. Round to six
decimal places as needed.
a. f (3)
b. f (1)
c. f (0)
d. f 1 12 2
a. f 132 ϭ e3 Ϸ 20.085537
c. f 102 ϭ e0 ϭ 1 (exactly)
b. f 112 ϭ e1 Ϸ 2.718282
1
d. f 1 12 2 ϭ e2 Ϸ 1.648721
Now try Exercises 31 through 36
C. You’ve just seen
how we can graph base-e
exponential functions
Although e is an irrational number, the graph
of y ϭ ex behaves in exactly the same way and has
the same characteristics as other exponential graphs.
Figure 5.21 shows this graph on the same grid as
y ϭ 2x and y ϭ 3x. As we might expect, all three
graphs are increasing, have an asymptote at y ϭ 0,
and contain the point (0, 1), with the graph of y ϭ ex
“between” the other two. The domain for all three
functions, as with all basic exponential functions, is
x ʦ 1Ϫq, q 2 with range y ʦ 10, q 2. The same
transformations applied earlier can also be applied to
the graph of y ϭ ex. See Exercises 37 through 42.
ᮣ
Figure 5.21
10
y y ϭ 3x
9
8
y ϭ 2x
7
6
5
y ϭ ex
4
3
2
1
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
Ϫ1
1
2
3
4
5
x
Ϫ2
D. Solving Exponential Equations Using
the Uniqueness Property
Since exponential functions are one-to-one, we can solve equations where each side is
an exponential term with the identical base. This is because one-to-oneness guarantees
a unique solution to the equation.
WORTHY OF NOTE
Exponential functions are very
different from the power functions
studied earlier. For power functions,
the base is variable and the
exponent is constant: y ϭ xb, while
for exponential functions the
exponent is a variable and the base
is constant: y ϭ bx.
Exponential Equations and the Uniqueness Property
For all real numbers m, n, and b, where b 7 0 and b
1.
If b ϭ b ,
then m ϭ n.
m
n
2.
1,
If m
then bm
Equal bases imply exponents are equal.
n,
bn