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Section 8.1
Exponential Functions
June 29, 2017
page 608
8.1 Exponential Functions
A
Examine differences between polynomial and exponential
functions.
B
Graph exponential functions.
Let’s look at a story that illustrates the difference between a polynomial and an
exponential function.
A
Examine differences between polynomial and exponential functions.
In an ancient story the king asks the inventor of the game of chess, “How can I
reward you?” The inventor says, “Put two pennies on the first square of the chess
board, four on the second, eight on the third, and so on, doubling the amount for
each square up to the sixty-fourth and last square on the board. Please reward me
by giving me the money on only the last square.”
The king, who knows that the inventor is a very clever person, asks himself, “Is this
some kind of trick?” So he reasons as follows. Does the inventor mean:
number of the square
1
2
3
4
5
...
64
number of pennies
2
4
6
8
10
...
128
x
2x
The king thinks that 128 pennies, or $1.28 is too small an amount. Does the
inventor mean square the amount on each square? Let’s see:
number of the square
1
2
3
4
5
number of pennies
1
4
9
16
25
...
64
... 4096
x
x2
The king thinks that’s more like it. "I can afford to reward the inventor with $40.96."
Since the king could only think of polynomials, so he agreed.
What the inventor had actually asked for can be described by the following table:
number of the square
1
2
3
4
5
...
number of pennies
2
4
8
16
32
...
21
22
23
24
25
...
64
264
x
2x
How much is 264 pennies? Using a calculator, we find that 264 pennies is about
$1.85 x 1017. That's 185 followed by 15 zeros, or 185 million billion dollars. How
much money is this? There are about 6 billion people in the world today. If we divide
Section 8.1
Exponential Functions
June 29, 2017
page 609
185 million billion dollars among each person in the world, each person would get
about 30 million dollars. Everyone would be a millionaire!
The king went broke trying to reward the inventor. But the point of this story is that
this new function f(x) = 2x ("2 raised to the exponent x", or "2 to the x") is very
different from the polynomial function g(x) = x2.
Example 1. Comparing f(x) = 2x and g(x) = x2
Compare the graphs of f(x) = 2x and g(x) = x2. Describe their differences.
Solution
First graph both f(x) = 2x and g(x) = x2 on the same coordinates, using an integer
window, as shown in
.
31
47
-47
-31
x
2
Figure 1. Graphs of f(x) = 2 and g(x) = x
At first glance, the curves look similar. Using a decimal window, though, we see that
the graph of f(x) = 2x hugs the x-axis, unlike the graph of g(x) = x2. See
.
y= x 2
3.1
y= 2x
-4.7
4.7
-3.1
x
2
Figure 2. Graphs of f(x) = 2 and g(x) = x
In the x-interval [0, 2], the two curves are close to each other, and meet when x is 2,
as shown in
.
Section 8.1
Exponential Functions
June 29, 2017
page 610
5
y= 2 x
0
2
0
y=x 2
x
2
Figure 3. Graphs of f(x) = 2 and g(x) = x
But in the interval (2,  ), the curves diverge, as shown in
.
x
y= 2
1000
y=x
2
20
2
0
x
2
Figure 4. Graphs of f(x) = 2 and g(x) = x

––––––––
x
The graph of f(x) = 2 in the example above shows that for x greater than 2, the value
2x begins to increase tremendously. If only the king had known!
The calculator graphs of the function f(x) = 2x show that the domain of f is the set of
real numbers. That is, 2x is valid for all real number exponent x. Real number
exponents follow rules and definitions of whole number exponents summarized
below.
Rules of Exponents
For a and b positive real numbers, a ≠ 1, b ≠ 1, and x and y
real numbers:
0
Zero Power:
Identity:
b 1
1
b b
Negative Power:
b
Product:
b b b
bx
x y
y b
b
Quotient:
Power:
x

x y
x
1
1 
x   
b
b
x y
bx   bxy
y
Section 8.1
Exponential Functions
June 29, 2017
page 611
ab x  a xb x
x
a 
ax
 x
b 
b
Example 2. Using rules of exponents
Use rules and definitions of exponents to rewrite the powers.
0
x
3
(a)
(b)
2
2 2
(c)
(d)
2
x
(e)
Solution
(a)
0
x
3
2 2  2
(c)
2x
x 3
3 2
2
2 x 
(f)
x
2 3
x
x3
Apply product rule.
Apply quotient rule.
1
2x
Definition of negative power
1 x
2

Apply power rule.
2x   23x
Apply power rule.
(d)

(e)
3
Definition of zero power
2 1
(b)
2x 
2x
23
3
2 x 3 x  2  3 x
(f)
6
Apply power rule.
x
––––––––
To graph f(x) = 2x by hand, we first build a table of values.
x
Example 3. Building a table of values of f(x) = 2
Compute a table of values of f(x) = 2x.
Solution
Starting with x = 0, use the definition a0 = 1, a ≠ 0 to get f 0  20  1.
x
Computing f(x) = 2 is easy for x a positive integer, for example, f 4  24  16 .
To compute f(x) = 2x for x a negative integer, use the definition of a negative power.
1
For example, f 4  24  16
.
 
We get the following table:
Section 8.1
x
f(x) = 2x
Exponential Functions
-100
-4
-3
1
2100
1
16
1
8
June 29, 2017
page 612
-2
-1
0
1
2
3
4
100
1
4
1
2
1
2
4
8
16
2
100
––––––––

Plotting some of the points displayed in the table, and connecting them with a
smooth curve gives the graph of the function f(x) = 2x.
Example 4. Graphing f(x) = 2x
Using the table of values given in Example 3, graph the function f(x) = 2x.
Solution
Plotting the points, as in
, we see a very distinct shape. By connecting the
dots, we get the graph of f(x) = 2x, as shown in
x
Figure 5. Plotted points satisfying f(x) = 2
.
x
Figure 6. Graph of f(x) = 2

––––––––
Let’s make some observations about the graph of y = 2x in
.
1. As x increases, so does y.
2. As x decreases to  y decreases nearly to zero (but never equals 0). The
x-axis is a horizontal asymptote of the graph y = 2x.
3. The graph passes through the point (0, 1).
4. The domain of f is , , and the range is 0,  .
The four representations of the exponential function f(x) = 2x are summarized below.
Section 8.1
Exponential Functions
Rule
Assign to the input
value x an output
value which is two
raised to the
exponent x.
June 29, 2017
Equation
Table
page 613
Graph
10
y = 2x
x
-3
y
1
8
-2
-1
0
1
2
3
1
4
1
2
1
2
4
8
(0, 1)
-5
5
The function f(x) = 2x is an example of an exponential function, defined below.
Definition
Exponential Function
The function
f(x) = bx, where b ≠ 1 is a positive real number,
is an exponential function.
Why is the exponential function f(x) = bx defined for b > 0, but b ≠ 1? The restriction
b ≠ 1 is necessary because f(x) = 1x is the constant function f(x) = 1, not an
exponential function.
The restriction b > 0 is necessary so that an exponential function has the set of real
numbers as its domain. For instance, if b = -2, then we have f(x) = (-2)x, where 1
2
(among infinitely many other real numbers) can't be included in the domain, since
 2 or
1
2

B

2 is not a real number.
Graph exponential functions.
Let’s look at some graphs of f(x) = bx, where b > 0, b ≠ 1.
Example 5. Graphing f(x) = bx, b > 1
Using a graphics calculator, graph f(x) = bx, for b = 1.1, 3, and 10. Describe any
similarities and differences.
Solution
20
-4.7
-1
20
-4.7
4.7
Figure 7. Graph of f(x) = 1.1
-1
x
20
4.7
x
Figure 8. Graph of f(x) = 3
-4.7
-1
4.7
x
Figure 9. Graph of f(x) = 10
, , and , we see that as x increases, so does y. The value 10x
grows faster than the value 3x, and 1.1x has the slowest growth of the three. The
By observing
Section 8.1
Exponential Functions
June 29, 2017
page 614
point (0, 1) is on each graph, since 1.10= 1, 30 = 1, and 100 = 1. For each graph, as x
decreases to -  y nears zero; thus, the line y = 0 (x-axis) is a horizontal asymptote.
The domain of each function is the set of real numbers; the range is (0,  ), the
positive real numbers. All three curves are continuous and smooth.
––––––––
Even though a calculator graph
pictures the curve touching the x-axis, the curve
actually just gets close to it. The x-axis is a
horizontal asymptote. Try tracing.
Functions such as f(x) = 1.1x, f(x) = 3x, and f(x) = 10x are examples of exponential
growth functions. Below is a summary of their behavior.
f(x) = bx, b > 1
Exponential Growth Function
1.
2.
3.
4.
5.
As x increases, so does y.
The x-axis is a horizontal asymptote of the graph of f.
The graph passes through the point (0, 1).
The domain of f is , , and the range is 0,  .
The curve is continuous and smooth.
Let’s now look at some graphs of f(x) = bx, where 0 < b < 1.
Example 6. Graphing f(x) = bx, 0 < b < 1
Using a graphics calculator, graph f(x) = bx, for b = 0.1,
similarities and differences.
Solution
20
-4.7
-1
and 0.8. Describe any
20
-4.7
4.7
Figure 10. Graph of f(x) = 0.1
By observing
1
3,
,
-1
x
20
1 
Figure 11. Graph of f(x) = 3
, and
-4.7
4.7
x
4.7
-1
x
Figure 12. Graph of f(x) = 0.8
, we see that for each graph, as x increases, the
x
value of y decreases. The value 0.1x decreases faster than the value 13 , and 0.8x

has the slowest decrease of the three.
Like the exponential growth functions, the point (0, 1) is on each graph, since
0
0.10= 1, 13  = 1, and 100 = 1. For each graph, as x increases to  , y nears zero, so
Section 8.1
Exponential Functions
June 29, 2017
page 615
the x-axis is a horizontal asymptote. The domain of each function is the set of real
numbers; the range is (0,  ). All three curves are continuous and smooth.
––––––––

x
Functions such as f(x) = 0.1x, f(x) = 1 , and f(x) = 0.8x are examples of exponential
3 
decay functions. Below is a summary of their behavior.
Exponential Decay Function
1.
2.
3.
4.
5.
f(x) = bx, 0 < b < 1.
As x increases, the value of y decreases.
The x-axis is a horizontal asymptote of the graph of f.
The graph passes through the point (0, 1).
The domain of f is , , and the range is 0,  .
The curve is continuous and smooth.
Notice that the only difference between exponential growth and exponential decay
functions is that the growth functions describe an increasing function, and decay
functions describe a decreasing function.
Exercises 8.1
A
Examine differences between polynomial and exponential functions.
1.
When Sally was 25 years old, she put away $2000 in a money market fund
which doubled her money every 8 years. Harry, on the other hand, decided to
put away $2000 in a fund that promised to give him 20 times the original
amount at the end of 40 years. When they retired 40 years later, how much
money did Sally have in the fund? How much did Harry have? Whose money
grew exponentially?
2.
Which method of investment would you choose, and why?
(1)
Money is doubled every 5 years.
(2)
Money is multiplied by the number of years it has been invested.
3.
Describe differences between linear and exponential functions. In your
description explain why you might use an exponential function, and not a
linear function, to model population growth. On the other hand, explain why
you would use a linear function, and not an exponential function, to model the
distance traveled at a constant rate of speed.
4.
Compare and contrast power and exponential functions. Illustrate with both
equation and graphic representations.
Section 8.1
Exponential Functions
June 29, 2017
page 616
Exercises 5 - 6. Use a graphics calculator to compare the graph of f(x) = 2x with
the graph of the given function g. Describe any differences.
5.
g(x) = 2x
6.
g(x) = 10x
Exercises 7 - 10. Use rules of exponents to rewrite the powers.
x
4
2
x
7.
(a) 3  3
8.
(a) 5  5
9.
(b)
34
3x
(c)
3x 
2
(a) 3
0
(b) 3
x
(c) 3
x1
10.
2
x 1
(b)
5x
52
(c)
5x 
3
0
(a) 7
(b) 7
x
(c) 7
x1
5
x 1
Exercises 11 - 12. List the letters corresponding to exponential functions.
u
1
1 
11. (a) f(x) =
12. (a) f(u) =
4 
x 7
(b) g(w) = 1.5w
(b) g(x) = 3x 6
(c) h(x) = 52 x
1
(d) l(t) = t
2
(c) h(x) = 4
1
(d) l(r) =
r 4
x
13.
Give the other three representations of the function whose rule is "Assign to the
input value x an output value which is four raised to exponent x."
14.
Give the other three representations of the function whose rule is "Assign to the
input value x an output value which is the reciprocal of four, raised to
exponent x."
Exercise 15 - 18. Match the function with its graph (a), (b), (c), or (d).
16. g(x) = x2
15. f(x) = 2x
17.
h(x) =
1
x
18.
j(x) = x3
Section 8.1
Exponential Functions
(a)
June 29, 2017
(b)
10
-4.7
4.7
10
-4.7
4.7
-10
(c)
page 617
-10
(d)
10
-4.7
4.7
10
-4.7
4.7
-10
-10
Exercises 19 - 22. Evaluate the given exponential function for the input values
-2, -1, 0, 1, and 2. Write your results in function notation and in table format.
19.
20. g x  5x
f x  3x
21.
B
g x  
x
1 
3 
22.
g x  
x
1 
5 
Graph an exponential function.
Exercises 23 - 26. Given the following three functions:
(i)
Graph each function, using a graphics calculator.
(iii)
Explain how the shape of the graph of y = bx changes, as the value of b
changes. Do the graphs show an exponential growth or an exponential
decay?
Determine the domain and the range of each function.
23.
y = 1.2x
y = 4x
y = 9x
24.
y1 = 1.9x
y2 = 5x
y3 = 7x
25.
y = 0.1x
y =
2
23 
y = 0.9x
26.
y1 = 0.2x
y2 =
25 
y3 = 0.8x
27.
What is the difference between exponential growth functions and exponential
decay functions?
(ii)
1
1
2
3
x
x
3
Section 8.1
Exponential Functions
June 29, 2017
page 618
28.
Explain why, in the definition of exponential function, the base b must be
positive.
29.
Describe similarities and differences between the following exponential, linear,
1
power, and rational functions: f(x) = 2x, l(x) = x, g(x) = x3, and h(x) =
.
x
30.
Describe similarities and differences between the following exponential, linear,
x
1 
power, and rational functions: f(x) =
, l(x) = -x, g(x) = -x3, and
2 
1
h(x) = .
x
Exercises 31 - 34. Translate the sentence into a graph that describes a
relationship between two variables. Label the horizontal and vertical axes
appropriately.
31. The total savings you are accumulating every year grows slowly at first, but
then increases rapidly.
32.
The population of a new community grows slowly during the first ten years, but
when people realize how good the schools are, it increases rapidly.
33.
As an automobile gets older, it loses its value rapidly at first, but after the first
few years, its value decreases more slowly.
34.
The Half-life of a radioactive isotope is 100 years. That is, after 100 years, 50%
of the original radioactive isotope remains. After 200 years, half of 50%, or
25% of the original isotope remains. After 300 years, half of 25%, or 12.5%
remains, and so on.
Section 8.1
35.
Exponential Functions
June 29, 2017
page 619
(A)
x
Each of the three functions, represented by a table and a graph, is increasing in
a different way. Match the table with its graph. And guess the function.
(I)
1
2
3
4
5
6
y
5
7
9
11
13
15
(B)
x
(II)
1
2
3
4
5
6
y
5
8
13
20
29
40
(C)
x
1
2
3
4
5
6
y
5
7
11
19
35
67
36.
Each of the three functions, represented by a table and a graph, is decreasing
in a different way. Match the table with its graph. And guess the function.
(I)
0
1
2
3
4
5
(A)
x
y
(B)
x
y
(C)
x
y
(III)
25
20
15
10
5
0
(II)
0
1
2
3
4
5
25
24
21
16
9
0
0
1
2
3
25
5
1
0.2
(III)
4
5
0.04 0.008
• Comprehensive and Continuation Exercises
Exercises 37 - 38. (i) Using a graphics calculator, graph f and g, for 0 ≤ x ≤ 10.
(ii) Determine the interval(s) for which f(x) is greater than g(x). (Round the
Section 8.1
Exponential Functions
June 29, 2017
page 620
endpoints of the interval(s) to one decimal place.) Hint: You may want to
consider the graph of f – g, since if f(x) – g(x) > 0, then f(x) > g(x).
37.
f(x) = 3x and g(x) = x3.
39.
Exponential decay function may also be defined as f x   a x , where
a > 1. Explain why this function is equivalent to the decay function given in
this textbook: f x   b x , where 0 < b < 1.
40.
Using a graphics calculator, graph f(x) = 2x + 2-x and g(x) = x2 + 2. Compare
and contrast the two functions.
41.
The exponential function f(x) = bx is defined for b > 0, b ≠ 1. To examine why
we have the restriction b > 0:
(i)
Graph y = (-1)x using an integer window. Describe what you see. Use
trace.
(ii)
Graph y = (-1)x with a decimal window. Describe what is graphed. Use
trace.
(iii) Why do you think b must be greater than 0?
38.
f(x) = 5x and g(x) = x5.