1.
Exponential Functions

2.
Objectives
To use the properties of exponents to:
 Simplify exponential expressions.
 Solve exponential equations.
To sketch graphs of exponential functions.

3.
Exponential Functions
A polynomial function has the basic form: f (x) = x3
An exponential function has the basic form: f (x) = 3x
An exponential function has the variable in the
exponent, not in the base.
General Form of an Exponential Function:
f (x) = Nx, N > 0

4.
Properties of Exponents
X Y
A A
  X Y
A 
XY
A
X Y
A 
 
Y
X
A 
X
Y
A
A

 
X
AB  X X
A B
X
A
B
 

 
 
X
X
A
B
X
A

1
X
A
1
X
A
 X
A
X
Y
A  Y X
A  
X
Y
A


5.
Properties of Exponents
2 3
2 2
 
5
2 32

2 6
2 2
  4
2
4
1
2

1
16

 
2
3
2  6
2 64

Simplify:

6.
Properties of Exponents
3
2
3

 

 
 
3
3
2
3


3
3
3
2

7
9
3
3

2
3
2
1
3

1
9

  
1 1
2 2
2 8 
1
2
16 16

27
8

4

 
1
2
2 8
 
Simplify:

7.
Exponential Equations
Solve: Solve:
5 125
x

3
5 5
x

3
x 
1
2
( 1)
7 7
x 


1
2
1
x   
1
2
x 

8.
Exponential Equations
8 4
x

 
3 2
2 2
x

3 2
x 
2
3
x 
3 2
2 2
x

8 2
x

 
3 1
2 2
x

3 1
x 
1
3
x 
3 1
2 2
x

Solve: Solve:

9.
Exponential Equations
1
3
27
x 
 
1
3
3
3
27
x 
19,683
x 
 
1
3 27
x

 
1
3 27
x


3
x
 
3
x  
3
3 3
x


Not considered an
exponential equation,
because the variable
is now in the base.
Solve: Solve:

10.
Exponential Equations
3
4
8
x 
 
4
3 4
3
3
4
8
x 
 
4
3
8
x 
 
4
2
x 
16
x 
Not considered an
exponential equation,
because the variable
is in the base.
Solve:

11.
Exponential Functions
General Form of an Exponential Function:
f (x) = Nx, N > 0
g(x) = 2x
x
2x
g(3) =
g(2) =
g(1) =
g(0) =
g(–1) =
g(–2) =
8
4
2
1 1
2 1
2

2
2
2
1
2
 1
4


12.
Exponential Functions
g(x) = 2x
2
0
x 3 2 1 0 -1 -2
g(x) 8 4 2 1 1/2 1/4

13.
Exponential Functions
g(x) = 2x

14.
Exponential Functions
h(x) = 3x
x
2
1
0
9
3
1
 
h x
–1
–2
1
3
1
9

15.
Exponential Functions
h(x) = 3x

16.
Exponential Functions
Exponential functions with
positive bases greater than
1 have graphs that are
increasing.
The function never crosses
the x-axis because there is
nothing we can plug in for x
that will yield a zero
answer.
The x-axis is a horizontal
asymptote.
h(x) = 3x (red)
g(x) = 2x (blue)

17.
Exponential Functions
A smaller base means the
graph rises more
gradually.
A larger base means the
graph rises more quickly.
Exponential functions will
not have negative bases.
h(x) = 3x (red)
g(x) = 2x (blue)

18.
Exponential Functions
   
1
2
x
j x 
Exponential functions with positive
bases less than 1 have graphs that are
decreasing.

19.
Properties of graphs of
exponential functions
Function and 1 to 1
y intercept is (0,1) and no x
intercept(s)
Domain is all real numbers
Range is {y|y>0}
Graph approaches but does not
touch x axis – x axis is asymptote
Growth or decay determined by base

20.
The Number e
e  2.71828169
A base often associated with exponential functions is:

21.
The Number e
Compute:  
1
0
lim 1 x
x
x


x
–.1
–.01
–.001
2.868
2.732
2.7196
 
1
1 x
x

 
1
0
lim 1 x
x
x



x
.1
.01
.001
2.5937
2.7048
2.7169
 
1
1 x
x

 
1
0
lim 1 x
x
x



2.71828169


22.
The Number e
Euler’s number
Leonhard Euler
(pronounced “oiler”)
Swiss mathematician
and physicist

23.
The Exponential Function
f (x) = ex

24.
Exponential Functions
x
2
1
0
4
2
1
 
j x
–1
–2
1
2
1
4
   
1
2
x
j x 

25.
Why study exponential functions?
Exponential functions are used in our real world
to measure growth, interest, and decay.
Growth obeys exponential functions.
Ex: rumors, human population, bacteria,
computer technology, nuclear chain reactions,
compound interest
Decay obeys exponential functions.
Ex: Carbon-14 dating, half-life, Newton’s Law of
Cooling