Exponential Functions - Math
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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