Exponential Functions
By
Dr. Julia Arnold and Ms. Karen Overman
using Tan’s 5th edition Applied Calculus for the
managerial , life, and social sciences text
The Exponential Function
The exponential function with base a is denoted
by f(x) = ax , where a > 0, a  1 and x represents
any real number.
Note: a is a positive constant and the variable x appears
in the exponent, which is why the function is given the name
exponential function.
Let’s look at some examples of exponential functions.
Example 1: Graph
x
2x
0
1
1
2
2
4
-1
1/2
-2
1/4
f x  2x using a table of values.
y
x
Let’s look at another example.
x
1
Example 2: Graph f x     using a table of values.
 3
x
X
1
 
 3
1
1/3
2
1/9
0
1
-1
3
-2
9
Now let’s look at some of the properties of exponential
functions.
1. The domain is all real numbers.
2. The range is (0, ). In other words for any x value the
function value is positive!!
3. The graph passes through the point (0, 1) since any
positive number raised to the zero power is 1.
4. The function is increasing if a > 1 and the function is
decreasing if 0 < a < 1.
Another exponential function, f(x) =ex is called
the natural exponential function.
e is called a transcendental number which
means it is not the root of another number.
e
anyroot
any number
On the graphing calculator, you can find e by pushing the yellow 2nd
function button and the LN key. On the display you will see e^( ,type
1 and close the parenthesis. You should see e^( 1). Press the enter key
and you will see 2.718281828 which represents an approximation of e.
To graph f(x) =ex , you will need to use your calculator.
Graph f(x) =ex using a table of values.
x
ex
1
e1 = 2.72
2 e2 = 7.39
0 e0 = 1
-1 e-1 = 0.37
-2 e-2 = 0.14
Let’s graph one more exponential for practice.
Graph f(x) = 4x .
First you should make a table of values. Then plot the points
and connect the points with a smooth curve.
x
4x
-2
-1
0
1
2
1/16
¼
1
4
16
Solving equations involving exponentials.
We are going to look at some equations involving exponential
functions. Before we do that let’s review the rules for
exponents.
Rules for exponents.
1. ax ay = ax+y
2. ax /ay = ax-y
3. (ax )y = axy
4. (ab)x = ax bx
5. (a/b)x = ax /bx
6. a0 = 1
7. a-n = 1/ an
Solving equations involving exponentials.
Example 1: Solve for x.
2x =32
Since, 32 can be written as 25, 2x = 25
and for the two sides to be equal their
exponents must be equal.
Thus, x = 5.
Solving equations involving exponentials.
Example 2: Solve for x.
9x = 1/27
Now, 27 is not 9 raised to a power, but both are powers of
3: 9 = 3² and 27 = 3³. Using rules 3 and 7 for exponents you
can rewrite each side so that each is 3 raised to a power.
9x = 1/27
Rewrite 9 and 27 as 3 to a power:
(3²)
Use the rules for exponents to simplify:
For the two sides to be
equal their exponents must
be equal.
3
x
= 1/(3³)
2x
= 3-3
Thus, 2x = -3 or x = -3/2.
Let’s review.
1. An exponential function has a positive constant base
(other than 1) and a variable exponent.
2. To graph an exponential you can use a table of values.
Then plot the points and connect them with a smooth
curve.
3. To solve an exponential equation, rewrite each side so
that they have the same base. Then set the exponents
equal and solve for the variable.
Note: In the next section we will solve more exponential
equations and we will look at the case where we cannot
rewrite the two sides so that they have the same base.