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 = x^{2} , 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 = x^{2} 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 = x^{2} , 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 TI83+/84+
graphing calculator
to investigate
inverses.
Click here. 

