Exponential
Functions
Functions of the form
are called exponential functions.
The domain of
is the set of all real numbers.
The range of
is the set of all positive numbers.
is called the base.
is required in order to avoid imaginary numbers.
For
example,
is imaginary.
Basic Exponential Graphs
The graph
of
The graph
of
These two graphs illustrate the basic shapes.
Basic properties
All graphs
1) go through the (0,1)
2) are continuous, no holes, no jumps
3) the x-axis is a horizontal asymptote
4) if
, the graph always goes up as a point
on the graph moves to the right
5) if ,
the graph always goes down as
a point on the graph moves to the right
6) the function
is one-to-one.
Property 6 means that the exponential function has an inverse
that is also a function.
We can solve the equation
for ,
.
This inverse function
is called the logarithmic function.
We soon will study it.
See Example 1, page 357.
Exponential Function Properties
The functions obey all the basic laws for exponents.
In addition,
if and only if f .
If , then
if and only
if .
These last two properties are useful in solving exponential equations.
See Example 2, page 358.
Applications of Exponential Functions
Population growth, radioactive decay, compound interest.
See Examples 3 - 5, pages 359 - 363.
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Exponential Function with Base e