# Inverse Functions

**Finding The Inverse of a Function**

In the beginning of this section when we defined functions by giving a table
of data we found the inverse by

interchanging the two columns of data.

If a function is described by an expression, to find the rule that defines a
function’s inverse we interchange the

x’s and the y’s in the expression, and then solve for y. For the inverse to be a
function we must guarantee that

the original function is one-to-one.

**Example 1: **Find f^{-1} for
Verify your results using the property of
inverse functions.

Support your conclusion graphically.

Solution:

We know that the graph of the line
passes the horizontal line test , and is therefore a

one-to-one function. To prove that f is one-to-one algebraically, we use the
definition of one-to-one to

show that
Therefore, f is one-to-one.

Write the function as

Interchange the x and the y:

Solve for y:

[The
resulting y is f^{-1}.]

•Verify your results by showing that

•Support graphically by showing that the graphs of f and f^{-1} have
symmetry about the line y=x.

**Example 2**: The function
is not a one-to-one function. Restrict the domain of f so that its

inverse will be a function.

Solution: Note from the graph of f, the part of the graph on either side of
the line x=0 is one-to-one.

Therefore, we will restrict the domain to be

Domain of f is

Range of f is

**Example 3: **In the above example we restricted the domain of
so that f has an inverse

function. Find f^{-1} and its domain and range.

Solution: The domain of f was restricted to x≥0, and the range is y≥-4 so that f is one-to-one.

•We write:

•Interchange x and y:

[Interchange domain and range.]

•Solve for y:

Since y≥0, we use only the positive root.

•Domain of f^{-1}:

Range of f^{-1}: