Topics
3. Relations & Functions
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United States of AmericaVirginia
Grade 8

3.03 Domain and range

Lesson
Practice

Domain and range

Domain and range describe the span of input and output values of a relation. Their definitions are very similar, but the small difference is very important.

The word function in a function machine box with domain being the input to the function machine and range being the output.
Domain

The set of all the input values for the independent variable or x-values (first number in an ordered pair)

Range

The set of all the output values for the dependent variable or y-values (second number in an ordered pair)

The domain and range of a discrete relation is represented as an ordered list of all of the unique values represented. Each of the following relations has a domain of \left\{-2,\,0,\,1,\,3\right\} and a range of \left\{-3,\,1,\,2\right\}:

A graph showing the ordered pairs (-2,1), (0,2), (1,1), and (3,-3). A set of the ordered pairs (3,-3),(0,1)(1,1), and (-2,2) in curly braces. An x-y table with (-2,2),(0,2),(0,1),(3,-3),(1,1),(0,-3),(-2,2) as pairs.

Even though all three relations have the same domain and range, only Relations 1 and 2 are functions. Keep in mind that the domain and range do not show us all of the input-output pairs, just all of the possible input and output values.

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 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.

This relation is not considered a function because the input 5 corresponds to two different outputs, 3 and -1. In a function, each input can correspond to only one output.

Example 2

Consider the relation in the table.

x1.5638.22
y32.8712.4
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}=\{1.5,\,6,\,3,\,8.2,\,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,\,2.8,\,7,\,1,\,2.4\}

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
x
2
4
6
8
10
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
x
2
4
6
8
10
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.

Reflect and check
2
4
6
8
10
x
2
4
6
8
10
y

By applying the vertical line test to the graph, we can see that a vertical line drawn at x=8 would intersect the graph at two points. This confirms that the relation is not a function since there's an x-value that is paired with more than one y-value.

Idea summary

Domain - a relation's inputs, or x-values

Range - a relation's outputs, or y-values

The domain and range of a discrete relation are typically written in ascending order without repeating values.

Outcomes

8.PFA.2

The student will determine whether a given relation is a function and determine the domain and range of a function.

8.PFA.2b

Identify the domain and range of a function represented as a set of ordered pairs, a table, or a graph of discrete points.

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