Download Sec 5.3 - Gordon State College

Sullivan Algebra and
Trigonometry: Section 5.3
Exponential Functions
Objectives of this Section
• Evaluate Exponential Functions
• Graph Exponential Functions
• Define the Number e
• Solve Exponential Equations
An exponential function is a function
of the form
f ( x)  a
x
where a is a positive real number (a >
0) and a  1. The domain of f is the
set of all real numbers.
Using a calculator to evaluate an
exponential function
Example: Find 2 1.41
On a scientific calculator:
2
yx
1.41
On a graphing calculator:
2
^
1.41
2 1.41 = 2.657371628...
The graph of a basic exponential function
can be readily obtain using point plotting.
(1, 6)
6
6x
3x
4
(1, 3)
(-1, 1/6)
3
(-1, 1/3)
2
(0, 1)
2
1
0
1
2
3
Summary of the Characteristics of the
x
graph of f ( x)  a , a  1
Domain: All real numbers
Range: (0,  )
No x-intercepts
y-intercept: (0,1)
Horizontal asymptote: y = 0 as x  
Increasing function
One-to-one
Summary of the Characteristics of the
x
graph of f ( x)  a , 0  a  1
Domain: All real numbers
Range: (0,  )
No x-intercepts
y-intercept: (0,1)
Horizontal asymptote: y = 0 as x  
Decreasing function
One-to-one
6
(-1, 6)
1

y 
 3
x
1

y 
 6
4
x
(-1, 3)
2
(0, 1)
3
2
1
(1, 1/3) (1, 1/6)
0
1
2
3
x
f
(
x
)

3
 2 and determine the
Graph
domain, range, and horizontal asymptote of f.
10
10
y  3 x
y  3x
5
5
(1, 3)
(0, 1)
0
(-1, 3)
(0, 1)
0
10
y  3 x  2
(-1, 5)
5
(0, 3)
y=2
0
Domain: All real numbers
Range: { y | y >2 } or (2,  )
Horizontal Asymptote: y = 2
The number e is defined as the number
that the expression
n
 1  1
 n
approaches as n  . In calculus, this is
expressed using limit notation as
n
1

e  lim  1  
n 
n
e  2.718281827
6
y3
x
ye
y2
4
2
3
2
1
0
1
2
3
x
x
Solve the following equations for x.
3
3
x 1
 27
x1
 3
3
x13
x4