lecture25 - 7 Asymptotes Lecture 25 8 Translations ex...

Lecture 25 1
Lecture 25: (Sec. 5.1)
Exponential Functions
Def. An Exponential function with base b
is the function defined by
where x is any real number, b > 0 and b 6 = 1.
How do we evaluate b
x
for all x ?

Lecture 25 2
ex. For f ( x ) = 4
x
, find:
1) f ( - 2)
2) f (
3
2
)
3) f (
√
3)
Be sure to notice the difference between these two
functions:
ex. f ( x ) = 4
x
ex. g ( x ) = x
4

Lecture 25 3
Laws of Exponents
Let x and y be any real numbers and let a and b be
positive. Then,
1) a
0
= 2) a
x
a
y
=
3)
a
x
a
y
= 4) ( a
x
)
y
=
5) ( ab )
x
= 6) (
a
b
)
x
=
7) a
- x
=

Lecture 25 4
ex. Simplify:
1) (
4
3
)
- 2
=
2)
25
-
2
3
5
25
1
6
3)
8
2 n +1
(
1
4
) n

Lecture 25 5
NOTE: We have another important property of
exponents.
8) Let x and y be real numbers and let
b > 0, b 6 = 1.
If b
x
= b
y
, then
NOTE: What happens if b = 1?
ex. Solve for x : (
1
27
)
2 x - 5
= 81
x

Lecture 25 6
The Graph of an Exponential
Function
ex. f ( x ) = 2
x
ex. f ( x ) = (
1
2
)
x
ex. f ( x ) = 2
- x

Lecture 25 7
Characteristics of Exponential
Functions
1) Domain: 1) Domain:
2) Range: 2) Range:
3) x -intercept(s): 3) x -intercept(s):
4) 4)
5) lim
x → + ∞
b
x
5) lim
x → + ∞
b
- x
6) lim
x →-∞
b
x
6) lim
x →-∞
b
- x
7) Asymptotes: