A
function and its inverse function can be
described as the "DO" and the "UNDO" functions. A function takes a
starting value, performs some operation on this value, and creates an
output answer. The inverse function takes the output
answer, performs some operation on it, and arrives back at the original
function's starting value.
This "DO" and
"UNDO" process can be stated as a composition of functions. If
functions f and g are inverse functions,
. A function
composed with its inverse function yields the original starting value.
Think of them as "undoing" one another and leaving you right where you
started.
So how do we find the inverse of a
function? 
Basically speaking, the process of
finding an inverse is simply the swapping of the x and y
coordinates. This newly formed inverse will be a relation, but may
not
necessarily be a function.
Consider this subtle difference in
terminology:
Definition: INVERSE OF A FUNCTION:
The relation formed when the independent variable is exchanged with
the dependent variable in a given relation. (This
inverse may NOT be a function.)
Definition:
INVERSE FUNCTION: If the above
mentioned inverse
of a function is itself a function, it is then called an
inverse function.

Remember: The
inverse of a function may not always be a function!
The original function must be a onetoone function
to guarantee that its inverse will also be a function. 
Definition: A function
is a onetoone function if and only if each
second element corresponds to one and only one first element.
(each x and y value is used
only once) Use the
horizontal line test
to determine if a function is a onetoone function.
If ANY horizontal line intersects your original function in ONLY ONE
location, your function will be a onetoone function and its inverse will
also be a function.
The function y = 3x + 2, shown at the right, IS a
onetoone function and its inverse will also be a function.
(Remember that the
vertical line test is used to show that a relation is a
function.)



Definition:
The inverse of
a function is the set of ordered pairs obtained by
interchanging the first and second elements of each pair in the
original function.
Should the
inverse of function
f (x) also be a function, this
inverse function is denoted by f
^{1}(x).
Note:
If the original function is a onetoone function, the
inverse will be a function. 
[The notation f ^{1}(x) refers to "inverse
function".
It does not algebraically mean 1/f (x).]
If a function is composed with
its inverse function,
the result is the starting value. Think of it as
the function and the inverse undoing one another when composed.
Consider the simple function f (x) = {(1,2),
(3,4), (5,6)}
and its inverse f^{ 1}(x) = {(2,1),
(4,3), (6,5)}

More specifically:
The answer is the starting value of 2. 
"So, how do we find
inverses?"
Consider the following three
solution methods:


Swap ordered
pairs: If your function is defined
as a list of ordered pairs, simply swap the x and y values.
Remember, the inverse relation will be a function only if the original
function is onetoone.
Examples:
a. 
Given function f,
find the inverse relation. Is the inverse relation also a function?
Answer:
Function f is a onetoone function since the x
and y values are used only once.
Since function f is a onetoone function, the inverse
relation is also
a function.
Therefore, the inverse function is:

b. 
Determine the inverse of this function. Is the inverse also a
function?
x 
1 
2 
1 
0 
2 
3 
4 
3 
f (x) 
2 
0 
3 
1 
1 
2 
5 
1 
Answer: Swap the
x and y variables to create the inverse relation.
The inverse relation will be the set of ordered pairs:
{(2,1), (0,2), (3,1), (1,0), (1,2), (2,3), (5,4),(1,3)}
Since
function f was not a onetoone function (the y
value of 1 was used twice), the inverse relation will NOT be a function
(because the x value of 1 now gets mapped to two separate y
values which is not possible for functions). 
Solve
algebraically:
Solving
for an inverse relation algebraically is a three step process:

1. Set the function = y
2. Swap the x and y variables
3. Solve for y 
Examples:
a. 
Find the
inverse of the function
Answer:

Remember:
Set = y.
Swap the variables.
Solve for y.Use the inverse function notation since
f (x) is a onetoone function. 

b. 
Find the inverse of
the function
(given that x is not equal to 0).
Answer:

Remember:
Set = y.Swap the variables.
Eliminate the fraction by multiplying each side by y.
Get the y's on one side of the equal sign by subtracting
y from each side.
Isolate the y by factoring out the y.
Solve for y.
Use the inverse function notation since f (x)
is a onetoone function.


Graph:
The graph of an inverse relation is the reflection of the
original graph over the identity line,
y = x. It may be
necessary to restrict the domain on certain functions to guarantee that
the inverse relation is also a function. (Read
more about graphing inverses.)
Example:
Consider the straight line,
y = 2x^{ }+ 3, as the original
function. It is drawn in
blue. If
reflected over the identity line, y = x, the original function
becomes the red dotted graph.
The new
red graph is also a straight line
and passes the vertical line test for functions. The inverse
relation
of y = 2x + 3 is also a function.
Not all graphs produce an inverse relation which is also a
function. 


Use the TI83+/84+
graphing calculator
to investigate
inverses.
Click here. 

