Inverse Functions

The inverse of a function undoes what the function does: it maps each output back to the original input. Inverse functions exist precisely when a function is one-to-one, and their graphs are reflections of each other across the line .

Quick Reference

Concept	Formula / Rule
Definition
Domain/range swap	;
Cancellation equations	and
When an inverse exists	If and only if is one-to-one
Graph reflection	Graph of is the reflection of graph of across
Warning

The Idea of Inversion

Consider . Given , we get . Now suppose we are told that and asked to find . We solve , giving .

In general, for any in the range of , the value of satisfying is given by . This defines as a function of , which we call . The function is the inverse of , since it undoes the effect of :

Definition

Suppose is a one-to-one function with domain and range . The inverse function is defined by

for every in .

Diagram showing the relationship between a function f and its inverse g, reversing the mapping between x and y. — The function and its inverse undo the effects of each other.

The domain and range of and simply swap:

Important notation warning: The in denotes an inverse function, not an exponent. In particular, .

Given that has an inverse and , , and , find , , and .

Solution

By the definition of inverse function:

Arrow diagram showing specific points mapping from domain to range via f. — Arrow diagram for : maps 1 to 3, 2 to −4, 5 to −1.

Arrow diagram showing specific points mapping from range to domain via f⁻¹. — Arrow diagram for : the mapping is exactly reversed.

Verifying an Inverse Function

Let be a one-to-one function with domain and range , and let be a function with domain and range . Then if and only if both of the following hold:

Proof

Suppose

. By definition,

. Substituting

for

gives

. Substituting

for

gives

. Both conditions follow. Conversely, suppose (1) and (2) hold. If

, then by (2),

, i.e.,

. If

, then by (1),

, i.e.,

. So

, which is the definition of

Writing the independent variable of as (rather than ), the cancellation equations become:

A function has an inverse function if and only if it is one-to-one. Equivalently, every increasing or decreasing (monotonic) function has an inverse function.

Verify that and are inverses of each other.

Solution

Both functions have domain

. We verify both cancellation equations:

Determine if each function has an inverse. (a) . (b) .

Solution

(a) We show

is one-to-one. If

is one-to-one and has an inverse. The graph of

is an upward shift of

and passes the horizontal line test.

Graph of y = x³ + 1 which is one-to-one. — passes the horizontal line test.

(b) Note that

, so

. Three distinct inputs give the same output, so

is not one-to-one and has no inverse.

Graph of y = x³ - x which is not one-to-one. — fails the horizontal line test.

How to Find the Inverse Function

Steps to find :

Write down the equation .
Solve for in terms of : .
The formula defines the inverse, with restricted to the range of .
If you prefer, replace every with to write .

Given , find its inverse function.

Solution

Set

and solve for

Therefore

Given , find .

Solution

Set

and solve:

. Replacing

with

Verification:

. ✓

Given , find .

Solution

Set

and solve:

, so

. The range of

, so:

Given with domain , find .

Solution

Set

and solve:

. Since

, we choose

. The range is

Graph of the Inverse Function

If is a point on the graph of , then is on the graph of . The point is the reflection of across the line .

The graphs of a function and its inverse are reflections of one another across the line .

Graph of the polynomial function f(x)=2x⁵ + 3 and its inverse reflected across the line y = x. — (a) and , reflected across .

Graph of the linear function g(x) = 2x - 1 and its inverse reflected across the line y = x. — (b) and , reflected across .

Graph of the function h(x)=√(2x+3) and its inverse reflected across the line y = x. — (c) and , , reflected across .

Given the graph of below, sketch the graph of .

Graph of a function with specific points labeled (2, 4), (3, 2), (4, 1), and (5, 0.5). — Graph of with points , , , labeled.

Solution

Reflect across

by swapping coordinates:

Solution graph showing the original function and its inverse plotted with reflected points. — The graphs of and are reflections of one another across .

Frequently Asked Questions

the same as

No. The notation

denotes the inverse function, not the reciprocal. For example, if

, then

, which is very different from

Does every function have an inverse?

No. A function has an inverse if and only if it is one-to-one. For instance,

does not have an inverse on all of

because

What does it mean that

and

are inverses of each other?

The relationship is symmetric: if

, then

. In other words,

and

undo each other in both directions.

Why does the graph of f⁻¹ look like a reflection of f across y = x?

Because the roles of

and

are swapped:

is on the graph of

exactly when

is on the graph of

. Swapping coordinates geometrically corresponds to reflecting across the diagonal line

How does restricting the domain allow us to define an inverse?

If a function is not one-to-one on its full domain, we can select a subdomain on which it is one-to-one. For example,

restricted to

is strictly decreasing and hence one-to-one, so it has an inverse on that restricted domain.

Quick Reference

The Idea of Inversion

Definition

Verifying an Inverse Function

How to Find the Inverse Function

Graph of the Inverse Function

Frequently Asked Questions

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