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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.

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: