Review of Exponential Functions
Example 1
Graph an Exponential Function
1
 2
x
Sketch the graph of y = 3   . Then state the function’s domain and range.
Make a table of values. Connect the points to sketch a smooth curve.
1
 2
x
x
y = 3 
–3
1
3 
 2
–2
1
3 
 2
–1
3 
0
1
3  = 3
 2
1
3   = 1.5
2
3   = 0.75
3
1
3   = 0.375
 2
1
 2
3
= 24
2
= 12
1
=6
0
1
1
 2
1
 2
2
3
The domain is all real numbers, and the range is all positive numbers.
Example 2
Identify Exponential Growth and Decay
Determine whether each function represents exponential growth or decay.
Function
1
6
Exponential Growth or Decay?
x
1
a. y = 2  
The function represents exponential decay, since the base,
b. y = 0.4(7)x
The function represents exponential growth, since the base, 7, is greater than 1.
6
, is between 0 and 1.
Example 3
Write an Exponential Function
Population In 1990, the population of Montana was 799,065 and it increased to 902,195 in 2000.
a. Write an exponential function of the form y = ab x that could be used to model the population y
of Montana. Write the function in terms of x, the number of years since 1990.
For 1990, the time x equals 0, and the initial population y is 799,065. Thus, the y-intercept, and value

of a is 799,065.
For 2000, the time x equals 2000-199 or 10, and the population y is 902,195. Substitute these values
and the value of a into an exponential function to approximate the value of b.
y = abx
902,195 = (799,065)b10
1.129  b10
10
1.129 = b
Exponential function
Replace x with 10, y with 902,195, and a with 799,065.
Divide each side by 799,065
th
Take the 10 root of each side.
To find the 10 th root of 1.129, use selection 5:


x
under the MATH menu on the TI-83/84 Plus.
KEYSTROKES: 10 MATH 5 1.129 ENTER 1.01220713
 growth of Montana from 1990 to 2000 is
An 
equation that models the population
x
y = 799,065(1.012) .
b. Suppose the population of Montana continues to increase at the same rate. Estimate the
population in 2010.

For 2010, the time equals 2010 – 1990 or 20.
y = 799,065(1.012) x
= 799,065(1.012) 20
1,014,361
Modeling equation
Replace x with 20.
Use a calculator.
 in Montana will be about 1,014,361 in 2010.
The population


Example 4
Solve Exponential Equations
Solve each equation.
a.
1
 
 2
1
 
 2
n–1
= 16
n –1
= 16
Original equation
(2–1)n – 1 = 24
Rewrite
1
–1
4
as 2 and 16 as 2 so each side as the same base.
2
Power of a Power
Property of Equality for Exponential Functions
Subtract 1 from each side.
Divide each side by –1.
2–n + 1 = 24
–n + 1 = 4
–n = 3
n = –3
1
 
 2
CHECK
1
 
 2
n –1
= 16
3 – 1
?
 16
1
 
 2
4
Original equation
Substitute –3 for n.
?
 16
Simplify.
16 = 16 
Simplify.
b. 55n + 1 = 125n – 2
55n + 1 = 125n – 2
55n + 1 = (53)n - 2
55n + 1 = 53n – 6
5n + 1 = 3n – 6
2n + 1 = –6
2n = –7
n = –3.5
Original equation
3
Rewrite 125 as 5 so each side has the same base.
Power of a Power
Property of Equality for Exponential Functions
Subtract 3n from each side.
Subtract 1 from each side.
Divide each side by 2.
55n + 1 = 125n – 2
55(-3.5) + 1 = 125(-3.5) – 2
5-16.5 = 125-5.5
0.000000000003 = 0.000000000003 
CHECK
Original equation
Substitute –3.5 for n.
Simplify.
Simplify.
Example 5
Solve Exponential Inequalities
Solve 9a – 5 < 729.
9a – 5 < 729
(32)a – 5 < 36
32a – 10 < 36
2a – 10 < 6
2a < 16
a<8
CHECK
Original inequality
2
6
Rewrite 9 as 3 and 729 as 3 so each side has the same base.
Power of a Power
Property of Inequality for Exponential Functions
Add 10 to each side.
 each side by2.
Divide
Test a value of a less than 8; for example, a = 5.
9a – 5 < 729
Original inequality
 729
Replace a with 5.
5–5
9
?
?
9  729
1 < 729 
0
Simplify.
Simplify.