Exponential Functions - Math

[Pages:11]Exponential Functions

In this chapter, will always be a positive number. a

For any positive number a > 0, there is a function f : R ! (0, 1) called

an

that is defined as ( ) = x.

exponential function

fx a

For example, ( ) = 3x is an exponential function, and ( ) = ( 4 )x is an

fx

gx

exponential function.

17

There is a big dierence between an exponential function and a polynomial.

The function ( ) = 3 is a polynomial. Here the "variable", , is being raised

px x

x

to some constant power. The function ( ) = 3x is an exponential function; fx

the variable is the exponent.

Rules for exponential functions

Here are some algebra rules for exponential functions that will be explained

in class.

If

n

2

N,

then

n

a

is

the

product

of

n

a's.

For

example,

34

=

3?3?3?3

=

81

0=1 a

If n, m 2 N, then

p

p

= n m m

n=(m

)n

a

a

a

1 x= a

x

a

The rules above were designed so that the following most important rule of exponential functions holds:

178

x y = x+y aa a

Another variant of the important rule above is

x

a = xy a

y

a

And there is also the following slightly related rule

( x)y = xy aa

Examples.

?

41 2

=

p 24

=

2

? 7 2 ? 76 ? 7 4 = 7 2+6 4 = 70 = 1

? 10

3=

1

3

=

1

10 1000

?

6

15

5

= 156

5 = 151 = 15

15

? (25)2 = 210 = 1024

?

(320)

1 10

=

32

=

9

?

8

2 3

=

1=

(8)

2 3

p1 =

( 3 8)2

1 22

=

1 4

*************

179

The base of an exponential function

If ( ) = x, then we call the of the exponential function. The base

fx a

a base

must always be positive.

Base 1

If ( ) is an exponential function whose base equals 1 ? that is if ( ) = 1x

fx

fx

?

then

for

n, m

2

N

we

have

pp

n

=

1n m

=

m 1n =

m1=1

f

m

In fact, for any real number , 1x = 1, so ( ) = 1x is the same function as

x

fx

the constant function f (x) = 1. For this reason, we usually don't talk much

about the exponential function whose base equals 1.

*************

Graphs of exponential functions

It's really important that you know the general shape of the graph of an exponential function. There are two options: either the base is greater than 1, or the base is less than 1 (but still positive).

Base greater than 1. If is greater than 1, then the graph of ( ) = x

a

fx a

grows taller as it moves to the right. To see this, let n 2 Z. We know

that 1 < a, and we know from our rules of inequalities that we can multiply

both sides of this inequality by a positive number. The positive number we'll

multiply by is n, so that we'll have a

n(1) n a ................
................

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