Graphically Represent the Inverse of a Function
Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

 

Definition of an Inverse Relation:
If the graph of a function contains a point (a, b), then the graph of the inverse relation of the function contains the point (b, a).  To graph the inverse of a function, reverse the ordered pairs of the original function.

Should the inverse relation of a function f (x) also be a function, this inverse function is denoted by f -1(x).

"The x- and y- coordinates
switch places!"

 

Note:  The inverse of a function MAY NOT, itself, be a function. 

If the inverse of a function is itself also a function, it is referred to as the inverse function.

 

Method 1:   Determine graphically if a function has an inverse which is also a function:

Use the horizontal line test to determine if a function has an inverse function.
If ANY horizontal line intersects your original function in ONLY ONE location, your function has an inverse which is also a function.

The function y = 3x + 2, shown at the right, HAS an inverse function because it passes the horizontal line test.

 

Method 2:  Determine graphically if a function has an inverse which is also a function:

If a function has an inverse function, the reflection of that original function in the identity line, y = x, will also be a function (it will pass the vertical line test for functions).

The example at the left shows the original function,
y = x2 , in blue.  Its reflection over the identity line
y = x is shown in red is its inverse relation.  The red dashed line will not pass the vertical line test for functions, thus y = x2 does not have an inverse function.
You can see that the inverse relation exists, but it is NOT a function.

NOTE:  With functions such as y = x2 , it is possible to restrict the domain to obtain an inverse function for a portion of the graph.  This means that you will be looking at only a selected section of the original graph that will pass the horizontal line test for the existence of an inverse function. 

For example:
 


or

} by restricting the graph in such a manner, you guarantee the existence of an inverse function for a portion of the graph.
(Other restrictions are also possible.)

 

The graph of :

The graph of a function composed with its inverse function is the identity line y = x.

 


Use the TI-83+/84+
graphing calculator
to investigate
inverses.
Click here.