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.


                       top
                       next      The Exponential Function with Base e