Exponential Functions
Exponential functions are used to model common real world situations, such as
population growth, compound interest, or even the decay of radioactive materials.
These functions represent rapid increases or declines which is best seen by viewing
graphs of the functions. The exponential function is defined as follows:
f(x) = ax where a is a positive number and x is any real number
We begin by evaluating functions for various values. Recall from our discussion on
functions that when we evaluate a function, we replace the variable in the function
with the value given in the function notation.
Example 1: Evaluate f (x) = 2x for f (3)
f (3) = 23
= 2∙2∙2 = 8
Example 2: Evaluate g(x) = 3x for g(2), g(0), g(-4)
g(2) = 32
= 3∙3 = 9
g(0) = 30
=1
g(-4) = 3-4
=
=
1
34
1
3∙3∙3∙3
=
1
81
Modified from College Algebra, by Carl Stitz, PhD and Jeff Zeager, PhD, CC-BY 2013. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
1 x
Example 3: Evaluate f (x) = ( ) for f (1), f (-1), f (-2)
2
1 x
( ) = (2-1)x = (2-x)
2
f (1) = 2-1
=
1
21
1
=
2
f (-1) = 2-(-1)
= 21 = 2
f (-2) = 2-(-2)
= 22 = 4
Example 4: Evaluate h(x) = 5x – 2 for h(0), h(1), h(2), h(3), h(-1)
h(0) = 50 – 2
= 5-2 =
1
52
=
1
25
h(1) = 51 – 2
= 5-1 =
1
5
h(2) = 52 - 2
= 50 = 1
h(3) = 53 – 2
= 51 = 5
h(-1) = 5-1 -2
= 5-3 =
1
53
=
1
125
To get a clearer understanding of exponential functions, it is important to graph the
function. We shall see that when a is greater than 1, the function will rise
dramatically. When a is less than 1, the function will fall dramatically.
Modified from College Algebra, by Carl Stitz, PhD and Jeff Zeager, PhD, CC-BY 2013. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 5: Graph: f (x) = y = 2x
Select values for x and solve for y:
1
1
1
x = -3
2-3 =
x = -2
2-2 =
x = -1
2-1 =
x=0
20 = 1
y=1
x=1
21 = 2
y=2
x=2
22 = 4
y=4
x=3
23 = 8
y=8
23
1
22
1
2
=
=
8
1
4
y=
y=
y=
8
1
4
1
2
Construct a table of values and graph:
x
-3
-2
-1
0
1
2
3
y
1
8
1
4
1
2
1
2
4
8
Notice that as x increases, the function values increase. Also as x decreases, the
function values get closer to zero but will never reach zero. The axis that the curve
gets closer to, in this case the x-axis, is called a horizontal asymptote.
Modified from College Algebra, by Carl Stitz, PhD and Jeff Zeager, PhD, CC-BY 2013. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
1 x
Example 6: Graph: g(x) = ( )
2
Select values for x and solve for y:
1 −2
x = -2
(2) = (2-1)-2 = 22 = 4
x = -1
(2) = (2-1)-1 = 21 = 2
x=0
(2) = 1
x=1
(2) = 2
x=2
(2) = (2-1)2 = 2-2 =
x=3
(2) = (2-1)3 = 2-3 =
1 −1
1 0
1 1
1 3
y=2
y=1
1
1 2
y=4
y=
1
4
1
8
y=
y=
1
2
1
4
1
8
Construct a table of values and graph:
x
-2
-1
0
1
2
3
y
4
2
1
1
2
1
4
1
8
Notice that as x decreases, the function values increase. Also as x increases, the
function values get closer to the horizontal asymptote.
Modified from College Algebra, by Carl Stitz, PhD and Jeff Zeager, PhD, CC-BY 2013. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 7: Graph: f(x) = 2x + 3
Select values for x and solve for y:
1
1
1
x = -4
2-4 + 3 = 2-1 =
x = -3
2-3 + 3 = 20 = 1
y=1
x = -2
2-2 + 3 = 21 = 2
y=2
x=0
20 + 3 = 23 = 8
y=8
x=1
21 + 3 = 24 = 16
y = 16
21
=
y=
2
2
Construct a table of values and graph:
x
-4
y
-3
-2
0
1
1
2
8
16
1
2
y= f(x) = 2x + 3
As x increases, the function values increase. As x decreases, the function values get
closer to the horizontal asymptote. Notice that the graph is similar to Example 5,
f (x) = 2x . However, this graph is shifted, or translated, three units to the left.
Modified from College Algebra, by Carl Stitz, PhD and Jeff Zeager, PhD, CC-BY 2013. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)