Inverse Functions


Recall the definition of when a relation, or equation, defines a function:
A relation between two variables - say,    and    - defines    as
a function of   , if to each value of    there corresponds at most
one value of  .
 
For example, the following equation defines as a function of

     ,
because to a given value of   , we can calculate only one value of  .
Given   , then  .

The equation    does not define    as a function of   ,
because to a given value of   , we can calculate not one but two values of  .
Given  ,
          
          .


Definition of One-to-One Function
A function is one-to-one if to each value of in its range,
there corresponds only one value of in its domain.
In other words, if a function is one-to-one, then the relation that we get
when we solve the relation for for ,
     
is also a function.
To a given value of   , there corresponds only one value of  .

The relation    is not one-to-one.

See Example 1, page 259 - 260.

Often it is not easy to determine if a function is one-to-one.
For example, it is not easy to solve the equation
     
for   in order to see whether one value of    determines only one value of  .
But if we know the graph of a function, we need only apply the

Horizontal Line Test.

          
                                              not one-to-one

If a horizontal line intersects the graph of a function in more than one point,
then the function is not one-to-one.

If each horizontal line intersects the graph of a function in at most one point,
then the function is one-to-one.

           

                                                 one-to-one

See Figure 1 - 2, pages 260 -261.


Finding the Inverse Function
If    is a one-to-one function, then it has an inverse function denoted by  .
If is not one-to-one, it does not have an inverse function.

If the function   is one-to-one, then we can solve the equation for
to get an equation expressing    in terms of   ; this equation expresses    as
a function of    and this function of    is called the inverse function of   ,
denoted by
      .

A little confusing, yes.
Here is an example illustrating the terminology and the notational conventions
employed.

Find the inverse function for  .

Recall that the equation can also be written as .
Recall that    and    both represent the ordinate of the same point
on the graph. The graph of and the graph of
are the same.
and both name the same point on the graph.

Okay, here is the process:
Use
     .
Solve this equation for .

     

       

Note that given one value for   , this equation gives only one value for   .

         is already the inverse function we are looking for.

The convention, however, is to use the   variable to write the inverse function.
Simply interchange the    and the   ,
           .

For inverse functions,    and   , it is true that
         
         

See Examples 2 - 3, pages 266 - 270.


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                                                    Module 4