Inverse Functions, Domain and Range

Ontario Grade 11 Mathematics

Characteristics of Functions

MCR3U study topic A1.6 — Understanding how inputs and outputs switch

A function takes an input and produces an output. Its inverse reverses that pairing: an output of the original function becomes an input of the inverse. This means the original function’s range becomes the inverse’s domain, and the original function’s domain becomes the inverse’s range. We will focus on this relationship, using finite sets and intervals to make the switch clear.

What you will learn

Explain what domain and range mean for a function.
Describe how the domain and range of a function relate to those of its inverse.
Use a table or a graph to recognize that an inverse switches inputs and outputs.

Prerequisite bridge: inputs, outputs, domain, and range

The domain is the set of allowed inputs to a function. The range is the set of outputs the function actually produces. For example, let

f(x)=2x+1

and use the domain

\{1,2,3\}

. The outputs are

3

,

5

, and

7

, so the range is

\{3,5,7\}

.

Function notation such as

f(2)

means the output when the input is

2

. In this example,

f(2)=5

. The notation helps us keep track of which values are inputs and which are outputs.

Domain means allowed inputs.
Range means actual outputs.
A function pairs each input with an output.

How an inverse switches the pairing

An inverse reverses the input-output pairing of a function. If the original function takes

a

to

b

, then its inverse takes

b

back to

a

. The symbol

f^{-1}

is read as “

f

inverse.” It names the reverse function; it does not mean the reciprocal of

f

.

The key relationship follows directly from this reversal. Every output of the original function is an input of its inverse, and every input of the original function is an output of its inverse. Therefore, the two sets switch roles.

This lesson concerns functions and their inverses. In each example, the inverse is given through the reversed pairings, so we can identify the domain and range without finding a new algebraic rule.

\operatorname{Domain}(f^{-1})=\operatorname{Range}(f),\qquad \operatorname{Range}(f^{-1})=\operatorname{Domain}(f)

If $f(a)=b$ , then $f^{-1}(b)=a$ .
The domain of the inverse is the range of the original function.
The range of the inverse is the domain of the original function.

See the switch in a table and on a graph

A table makes the reversal easy to see. Each row records one input-output pair. For the inverse, the original output becomes the new input, and the original input becomes the new output. As a result, the values in the domain and range exchange places.

The same idea appears on a graph. A point with coordinates

(x,y)

on the original function corresponds to a point with coordinates

(y,x)

on its inverse. Swapping the coordinates reflects the graph across the line

y=x

. The horizontal values of one graph become vertical values of the other.

(x,y)\longleftrightarrow(y,x)

Reversing each ordered pair switches its two coordinates.
The graph relationship reflects the domain-range switch.

Use the relationship carefully

To state the inverse’s domain and range, first identify the original function’s domain and range. Then exchange those sets. This method works whether the sets are listed values, such as

\{2,4,6\}

, or intervals, such as

[1,5]

.

Do not assume the domain and range stay in the same roles just because the same numbers appear in both sets. Read each set as inputs or outputs for the function being discussed. A set that is the range of the original function is the domain of its inverse.

Identify both original sets before switching them.
Keep the set values and their roles distinct.

The domain-range switch

Function	Domain	Range
$f$	Original inputs	Original outputs
$f^{-1}$	Original outputs	Original inputs

Worked example

Example 1 — A function given by a table

A function

p

has the input-output pairs shown below. State the domain and range of

p

, then state the domain and range of its inverse.

Read the original inputs
The inputs listed in the first row form the domain of $p$ .
$\{0,1,2,3\}$
Read the original outputs
The outputs listed in the second row form the range of $p$ .
$\{4,7,10,13\}$
Switch the two sets
The inverse reverses every input-output pairing. Therefore, the outputs of $p$ become inputs of $p^{-1}$ , and the inputs of $p$ become outputs of $p^{-1}$ .
$\operatorname{Domain}(p^{-1})=\{4,7,10,13\},\qquad \operatorname{Range}(p^{-1})=\{0,1,2,3\}$

Answer: The domain of

p

is

\{0,1,2,3\}

and its range is

\{4,7,10,13\}

. The domain of

p^{-1}

is

\{4,7,10,13\}

and its range is

\{0,1,2,3\}

.

Check: For instance, if

p(2)=10

, the reversed pairing is

p^{-1}(10)=2

. This confirms that an original output becomes an inverse input.

Worked example

Example 2 — A function with interval domain and range

A function

q

has domain

[1,5]

and range

[-2,6]

. State the domain and range of

q^{-1}

.

Identify the original sets
The given domain lists the inputs of $q$ , and the given range lists its outputs.
$\operatorname{Domain}(q)=[1,5],\qquad \operatorname{Range}(q)=[-2,6]$
Use the inverse relationship
The inverse takes outputs of $q$ as its inputs, so its domain is the original range. It returns original inputs, so its range is the original domain.
$\operatorname{Domain}(q^{-1})=[-2,6],\qquad \operatorname{Range}(q^{-1})=[1,5]$

Answer: The domain of

q^{-1}

is

[-2,6]

, and its range is

[1,5]

.

Check: The inverse domain matches the given range, and the inverse range matches the given domain.

Common mistakes and how to avoid them

Keeping the original domain as the inverse’s domain.

Correction: The inverse’s domain is the original function’s range. The original domain becomes the inverse’s range.

Treating

f^{-1}

as the reciprocal of the function.

Correction:

f^{-1}

names the inverse function, which reverses input-output pairs. It does not mean

1/f(x)

.

Switching the numbers but not their roles.

Correction: Label each set as domain or range for each function. The values that were outputs of the original become inputs of the inverse.

Lesson summary

The domain is the set of inputs, and the range is the set of actual outputs.
An inverse reverses the input-output pairings of a function.
The domain of the inverse is the range of the original function.
The range of the inverse is the domain of the original function.
On a graph, corresponding points switch coordinates.

Check your understanding

Question 1

A function

r

has domain

\{2,5,8\}

and range

\{-1,3,7\}

. What is the domain of

r^{-1}

?

$\{2,5,8\}$
$\{-1,3,7\}$
$\{-1,2,3,5,7,8\}$
correctIndex: 1

Show answer and explanation

\{-1,3,7\}

The inverse’s domain is the range of the original function, so it is

\{-1,3,7\}

.

Question 2

A point

(4,-2)

belongs to the graph of a function. What is the corresponding point on the graph of its inverse?

$(-2,4)$
$(4,2)$
$(-4,-2)$
correctIndex: 0

Show answer and explanation

(-2,4)

An inverse switches the input and output coordinates, so

(4,-2)

becomes

(-2,4)

.

Question 3

A function

s

has domain

[0,3]

and range

[2,9]

. What is the range of

s^{-1}

?

$[2,9]$
$[0,3]$
$[0,9]$
correctIndex: 1

Show answer and explanation

[0,3]

The inverse’s range is the original function’s domain, which is

[0,3]

.

Key terms

Domain: The set of allowed input values for a function.
Range: The set of output values a function actually produces.
Inverse function: A function that reverses the input-output pairings of another function.
Ordered pair: A pair of coordinates written in order as $(x,y)$ , representing an input and its output on a graph.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A1.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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A1.6 · Inverse Functions, Domain and Range

Ontario Grade 11 Mathematics

MCR3U study topic A1.6 — Understanding how inputs and outputs switch

What you will learn

Prerequisite bridge: inputs, outputs, domain, and range

How an inverse switches the pairing

See the switch in a table and on a graph

Use the relationship carefully

The domain-range switch

Example 1 — A function given by a table

Example 2 — A function with interval domain and range

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCR3U

Tutor support for A1.6 · Relate the domain and range of a function and its inverse

About this lesson and its review