Concept of a Relation: Domain, Range and Image

A relation is a way of connecting members of one set to members of another set. In Mathematics, relations are often shown using ordered pairs, tables, arrow diagrams, or rules. Learning about relations helps you understand the important ideas of domain, co-domain, range, and image. These ideas are also the foundation for studying functions.

What you will learn

• Explain what a relation is
• Explain why the order of an ordered pair matters
• Represent a relation using ordered pairs, a table, an arrow diagram, and a rule
• Read a relation from an arrow diagram
• Identify the domain, co-domain, range, and image
• Tell the difference between the range and the co-domain
• Recognise that one input can have more than one output, and that two inputs can share the same output
• Recognise examples and non-examples of relations

Key Vocabulary

Term	Meaning
Relation	A rule or connection that pairs members of one set with members of another set.
Input	The starting value in a relation.
Output	The value that an input is paired with.
Domain	The set of all input values.
Co-domain	The set of possible output values.
Range	The set of output values actually used by the relation.
Image	The output value that a particular input maps to.

Key Ideas

Understanding Relations

A relation can be written as a set of ordered pairs. For example:

This means:

• 1 is paired with 2
• 2 is paired with 4
• 3 is paired with 6

In each ordered pair, the first number is the input and the second number is the output.

Ordered Pair	Input	Output
(1, 2)	1	2
(2, 4)	2	4
(3, 6)	3	6

The Order Matters

The word ordered in ordered pair is important. The input always comes first and the output always comes second. Changing the order changes the relation.

Relation	Inputs (domain)	Outputs (range)
{(1, 2), (2, 4)}	{1, 2}	{2, 4}
{(2, 1), (4, 2)}	{2, 4}	{1, 2}

Arrow Diagram

An arrow diagram is a useful way to show a relation. The arrows show which input is paired with which output.

Domain, Co-domain and Range

Consider this relation:

Part	Answer	Reason
Domain	{1, 2, 3}	These are the input values.
Co-domain	{2, 4, 6, 8}	These are all the possible output values in set Y.
Range	{2, 4, 6}	These are the output values actually used.

Image of an Input

The image of an input is the output value that the input maps to.

For the relation:

• The image of 1 is 2.
• The image of 2 is 4.
• The image of 3 is 6.

Different Pairings You May See

So far, every input had its own output. But a relation can be built in other ways too. It only needs to pair values together.

What happens	Example	Domain	Range
One input is paired with more than one output	{(1, 2), (1, 3)}	{1}	{2, 3}
Two inputs are paired with the same output	{(1, 5), (2, 5)}	{1, 2}	{5}

Different Ways to Show a Relation

The same relation can be shown in different forms.

Form	Example
Ordered pairs	{(1, 2), (2, 4), (3, 6)}
Table	x 1 2 3 y 2 4 6
Arrow diagram	1 → 2, 2 → 4, 3 → 6 (see the diagram above)
Rule	y = 2x
Words	The output is twice the input.

Worked Examples

Example 1: State the domain and range of the relation {(1, 2), (2, 4), (3, 6)}.

1. Look at the first numbers in each ordered pair.
2. The first numbers are 1, 2, and 3.
3. So the domain is {1, 2, 3}.
4. Look at the second numbers in each ordered pair.
5. The second numbers are 2, 4, and 6.
6. So the range is {2, 4, 6}.

Answer: Domain = {1, 2, 3}, Range = {2, 4, 6}.

Example 2: For the relation {(0, 1), (1, 3), (2, 5), (3, 7)}, state the image of 2.

1. Find the ordered pair that begins with 2.
2. The ordered pair is (2, 5).
3. This means 2 maps to 5.

Answer: The image of 2 is 5.

Example 3: A relation is shown in the table below. State the domain and range.

x	2	4	6	8
y	5	7	9	11

1. The input values are in the x row.
2. So the domain is {2, 4, 6, 8}.
3. The output values are in the y row.
4. So the range is {5, 7, 9, 11}.

Answer: Domain = {2, 4, 6, 8}, Range = {5, 7, 9, 11}.

Example 4: The arrow diagram shows a relation from X to Y. State the ordered pairs, the domain, the co-domain, and the range.

1. Follow each arrow. 1 goes to 3, 2 goes to 5, 3 goes to 7, and 4 goes to 9.
2. The ordered pairs are (1, 3), (2, 5), (3, 7), and (4, 9).
3. The inputs are the elements of X that have arrows: 1, 2, 3, and 4. So the domain is {1, 2, 3, 4}.
4. The co-domain is all the elements of set Y: {1, 3, 5, 7, 9}.
5. The range is only the elements of Y that an arrow points to: 3, 5, 7, and 9.
6. The value 1 is in the co-domain, but no arrow points to it. So 1 is not in the range.

Answer: Ordered pairs = {(1, 3), (2, 5), (3, 7), (4, 9)}. Domain = {1, 2, 3, 4}. Co-domain = {1, 3, 5, 7, 9}. Range = {3, 5, 7, 9}.

Example 5: The relation is given by the rule y = x + 3. If the domain is {1, 2, 3}, find the range.

1. Substitute each input into the rule.
2. When x = 1, y = 1 + 3 = 4.
3. When x = 2, y = 2 + 3 = 5.
4. When x = 3, y = 3 + 3 = 6.
5. The outputs are 4, 5, and 6.

Answer: Range = {4, 5, 6}.

Example 6: Four students each name their favourite fruit. Kemar chooses mango, Shanice chooses banana, Andre chooses mango, and Tricia chooses guava. Write this relation as ordered pairs and state the domain and range.

1. Each student is the input and their favourite fruit is the output.
2. The ordered pairs are (Kemar, mango), (Shanice, banana), (Andre, mango), and (Tricia, guava).
3. The domain is the set of students: {Kemar, Shanice, Andre, Tricia}.
4. The range is the set of fruits chosen: mango, banana, and guava.
5. Mango was chosen by two students, but it is written only once in the range.

Answer: Domain = {Kemar, Shanice, Andre, Tricia}. Range = {mango, banana, guava}.

Examples and Non-Examples of Relations

A relation must show a clear connection or pairing between values. It may be shown by ordered pairs, a table, a diagram, or a rule.

Example	Is it a relation?	Reason
{(1, 3), (2, 5), (3, 7)}	Yes	It pairs each input with an output.
y = 4x	Yes	The rule connects each input x to an output y.
A table showing students and their ages	Yes	Each student is paired with an age.
{(1, 2), (1, 3)}	Yes	One input can be paired with more than one output.
{(1, 5), (2, 5)}	Yes	Two inputs can be paired with the same output.
{2, 4, 6, 8} only	No	This is only a set of numbers. It does not show any pairing.
A list of four students’ names on its own	No	The names are not paired with anything, such as an age or a fruit.

Common Mistakes

Multiple Choice Questions

Practice Questions

1. For the relation {(0, 1), (1, 3), (2, 5)}, state the domain.
2. For the same relation, state the range.
3. What is the image of 1?
4. For the relation {(2, 10), (4, 20), (6, 30)}, state the domain.
5. For the same relation, state the range.
6. What is the image of 6?
7. The co-domain is {3, 6, 9, 12} and the relation is {(1, 3), (2, 6), (3, 9)}. State the range.
8. In the previous question, which value is in the co-domain but not in the range?
9. A relation is given by y = 2x + 1. If the domain is {1, 2, 3}, find the range.
10. In your own words, explain the difference between the co-domain and the range.
11. The arrow diagram shows a relation from X to Y.
    (a) Write the relation as ordered pairs.
    (b) State the domain, the co-domain, and the range.
12. Which of the following are relations? Give a reason for each.
    (a) {(1, 4), (2, 5), (3, 6)}
    (b) {7, 8, 9}
    (c) the rule y = x − 2
    (d) a list of the names of four students on its own
13. Complete the table for the rule y = 3x, and then state the range.

    x	1	2	3	4
    y

14. Kemar likes mango, Shanice likes guava, and Andre likes mango. Write this relation as ordered pairs, and state the domain and the range.
15. Are (3, 7) and (7, 3) the same ordered pair? Explain your answer.

Answers

1. {0, 1, 2}
2. {1, 3, 5}
3. 3
4. {2, 4, 6}
5. {10, 20, 30}
6. 30
7. {3, 6, 9}
8. 12
9. When x = 1, y = 2(1) + 1 = 3.
   When x = 2, y = 2(2) + 1 = 5.
   When x = 3, y = 2(3) + 1 = 7.
   Range = {3, 5, 7}.
10. The co-domain is the set of possible output values, while the range is the set of output values actually used by the relation.
11. (a) {(2, 1), (4, 2), (6, 3)}
    (b) Domain = {2, 4, 6}. Co-domain = {1, 2, 3, 4}. Range = {1, 2, 3}. (The value 4 is in the co-domain but no arrow points to it.)
12. (a) Yes. It pairs each input with an output.
    (b) No. It is only a set of numbers, with no pairing.
    (c) Yes. The rule pairs each input x with an output y.
    (d) No. The names are not paired with anything.
13. When x = 1, y = 3(1) = 3.
    When x = 2, y = 3(2) = 6.
    When x = 3, y = 3(3) = 9.
    When x = 4, y = 3(4) = 12.
    The table shows y = 3, 6, 9, 12. Range = {3, 6, 9, 12}.
14. Ordered pairs: {(Kemar, mango), (Shanice, guava), (Andre, mango)}.
    Domain = {Kemar, Shanice, Andre}. Range = {mango, guava}. (Mango is written only once.)
15. No. In (3, 7) the input is 3 and the output is 7. In (7, 3) the input is 7 and the output is 3. The order matters, so they are different ordered pairs.

Lesson Summary