Exponential Functions
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Definition:
The exponential function with base b is defined by
                                    
             



Let's examine the function

 
Check out the table
and the graph.

x f (x) = y
-2
-1
0
1
2
3

Most exponential graphs resemble this same shape.
This graph is very, very small on its left side and is extremely close to the x-axis.  As the graph progresses to the right, it starts to grow faster and faster and shoots off the top of the graph very quickly, as seen at the right.

In a straight line, the "rate of change" remains the same across the graph.  In these graphs, the "rate of change" increases or decreases across the graphs.
 

Characteristics:
     Such exponential graphs of the form 
f (x) = bx  have certain characteristics in common:

    Exponential functions are one-to-one functions.
    graph crosses the y-axis at (0,1)

  when b > 1, the graph increases

  when 0 < b < 1, the graph decreases

  the domain is all real numbers

  the range is all positive real numbers (never zero)

  graph passes the vertical line test - it is a function

  graph passes the horizontal line test - its inverse is also a function.

  graph is asymptotic to the x-axis - gets very, very close to the x-axis but does not touch it or cross it.
 


 
Natural Exponential Function:

The function defined by  f (x) = ex  is called the natural exponential function.

(e is an irrational number, approximately 2.71828183, named after the 18th century Swiss mathematician, Leonhard Euler .)

Notice how the characteristics of this graph are similar to those seen above.

 This function is simply a "version" of
   where b >1.

  

Inverse of  f (x) = ex:

Since  f (x) = ex is a one-to-one function, we
know that its inverse will also be a function. 

But what is the equation of the inverse of
 
f (x) = ex ?
To solve for an inverse algebraically:

set the equation equal to y

swap the x and y

solve for y by rewriting in log form

log base e is called the natural log, ln x.

The inverses for other exponential functions
are found in this same manner.

 

Graphing the inverse of the natural exponential function by reflecting it over the identity function,
y = x, shows the natural logarithmic function,
y = ln x.
Notice how (0,1) from  f (x) = ex becomes (1,0)
for y = ln x.  The coordinates switch places
between a graph and its inverse.

Other Exponential Inverses:

Graph Inverse

set the equation equal to y

swap the x and y

solve for y by rewriting in log form

Graph Inverse

set the equation equal to y

swap the x and y

solve for y by rewriting in log form

Another view of finding the inverse algebraically:


Set the equation equal to y.

Swap the x and y variables.

Take the log of both sides.

Apply the rule log ar = r log a.

Solve for y.

Apply the change of base formula:
    logb a = log a / log b

 

How to use your
TI-83+/84+ graphing calculator with exponentials.
Click calculator.