Topics
3. Relations and Functions
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United States of AmericaVA
Grade 8

3.04 Domain and range

Lesson
Practice

Introduction

We have explored relations and functions as ways of linking two sets of data, usually written using x's and y's as coordinates.

We'll now consider all the possible values of inputs and outputs that these relationships can have.

Domain and range

Domain and range both describe the span of values that a relation can take. Their definitions are very similar, but the small difference is very important.

Domain

all of the possible x-values of a relation

Range

all of the possible y-values of a relation

We can find the domain and range of a relation no matter how it is represented. We simply need to look at the ordered pairs, graph, table, or other representation and list out all of the possible x and y-values that the relation can have, putting commas in between each value and curly braces on the outside. Do not repeat any values that show up more than once.

Typically the domain and range are written in ascending order (least to greatest) but that is not a requirement.

Examples

Example 1

Consider the relation \{(1, 2), (5, 3), (2, 7), (5, -1)\}. State the domain and range.

Worked Solution
Create a strategy

Look at the x-coordinates to determine the domain, then look at the y-coordinates for the range.

Apply the idea

The domain is \{1, 2, 5\} and the range is \{-1, 2, 3, 7\}.

Reflect and check

Notice the domain and range are written in ascending order (least to greatest). This helps keep things organized but it is not a requirement.

Also notice that the value of 5 was only included once in the domain. That is because we are only concerned with what all of the possible x values are and not how many times they showed up.

Example 2

Consider the relation in the table.

x16382
y32712
a

What is the domain of the relation?

Worked Solution
Create a strategy

Write all the x-values of the relation.

Apply the idea

\text{Domain}=\{1, 6, 3, 8, 2\}

b

What is the range of the relation?

Worked Solution
Create a strategy

Write all the y-values of the relation.

Apply the idea

\text{Range}=\{3, 2, 7, 1\}

c

Is this relation a function?

Worked Solution
Create a strategy

A relation is a function if every x-value in the domain has only one corresponding y-value in the range.

Apply the idea

The x-value in the domain has only one corresponding y-value in the range. So, this relation is a function.

Example 3

Consider the relation on the graph below.

2
4
6
8
10
12
x
2
4
6
8
10
12
y
a

What is the domain of the relation?

Worked Solution
Create a strategy

Write all the unique x-values of the points.

Apply the idea

First, identify the coordinates of the points in the graph:

2
4
6
8
10
12
x
2
4
6
8
10
12
y

Then write the unique x-values:

\text{Domain}=\{2, 4, 6, 8\}

b

What is the range of the relation?

Worked Solution
Create a strategy

Write all the unique y-values of the points from the graph in part (a).

Apply the idea

\text{Range}=\{2, 4, 6, 8, 10\}

c

Is this relation a function?

Worked Solution
Create a strategy

A relation is a function if every x-value in the domain has only one corresponding y-value in the range.

Apply the idea

The x-value of 8 in the domain has two corresponding y-values of 8 and 10 in the range. So, this relation is not a function.

Idea summary

Domain - all of the possible x-values of a relation

Range - all of the possible y-values of a relation

Outcomes

8.15b

Determine the domain and range of a function

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