A1.1 Distinguish functions from non-functions using multiple representations

Ontario Grade 11 Mathematics

Characteristics of Functions

Distinguishing Functions from Non-Functions Using Multiple Representations

Every time you type a value into a calculator and press a button, you expect exactly one answer. That idea — one input, one output — is the heart of what mathematicians call a function. Before you can work with functions in MCR3U, you need to be able to look at any representation of a relation and confidently decide whether it is a function or not. This lesson builds that skill using four different representations: ordered pairs, tables of values, mapping diagrams, and graphs. Each representation tells the same story in a different way, and you will learn to read all of them.

What you will learn

Explain what makes a relation a function in plain language.
Apply the vertical line test to graphs to decide whether a relation is a function.
Use sets of ordered pairs, tables, and mapping diagrams to identify functions and non-functions.
Explain why the same input cannot produce two different outputs in a function.

Prerequisite Bridge: Relations and Ordered Pairs

From Grade 9 and 10, you know that a relation is any pairing of values from one set (the inputs) with values from another set (the outputs). Each pairing is written as an ordered pair

(x, y)

, where

x

is the input and

y

is the output. For example, the set

\{(1, 3),\ (2, 5),\ (3, 7)\}

is a relation. The collection of all inputs is called the domain, and the collection of all outputs is called the range.

A relation does not have to follow any particular rule — it is simply a set of pairings. A function, however, is a special kind of relation that obeys one strict rule. Understanding that rule is the entire goal of this lesson.

A relation pairs inputs with outputs.
Each input-output pair is written as $(x, y)$ .
The domain is the set of all inputs; the range is the set of all outputs.
A function is a relation with an extra rule.

The Definition of a Function

A relation is a function if and only if every input in the domain is paired with exactly one output in the range. In other words, no single

x

-value is allowed to produce two or more different

y

-values.

Think of a function like a vending machine. You press one button (one input) and you get exactly one snack (one output). If pressing the same button sometimes gave you chips and other times gave you a chocolate bar, the machine would be broken — and the relation would not be a function.

Notice that the rule only restricts inputs. Two different inputs are perfectly allowed to share the same output. For example, both

x = -2

and

x = 2

can give

y = 4

. That is fine. What is not allowed is one input, say

x = 3

, giving both

y = 7

and

y = 9

at the same time.

One useful way to state the rule: each

x

-value appears at most once as a first element in the set of ordered pairs.

Every input must map to exactly one output.
Two different inputs may share the same output — that is allowed.
One input producing two outputs violates the function rule.
The key question to ask: does any $x$ -value repeat with a different $y$ -value?

Testing Ordered Pairs and Tables of Values

When a relation is given as a list of ordered pairs or as a table, check whether any

x

-value appears more than once with a different

y

-value. If it does, the relation is not a function. If every

x

-value is unique (or repeats only with the exact same

y

-value), the relation is a function.

Consider Relation A:

\{(2, 5),\ (4, 9),\ (6, 5),\ (8, 11)\}

. Every

x

-value — CAD 2, 4, 6, 8 — appears exactly once. Even though

y = 5

appears twice (for

x = 2

and

x = 6

), no single input repeats with a different output. Relation A is a function.

Now consider Relation B:

\{(3, 1),\ (5, 7),\ (3, -4),\ (9, 2)\}

. The input

x = 3

appears twice — first paired with

y = 1

and then with

y = -4

. Because one input leads to two different outputs, Relation B is not a function.

The same logic applies to a table of values: scan the input column. If any value repeats with a different output in the output column, the relation is not a function.

Scan the $x$ -values (inputs) for repeats.
A repeated $x$ with the same $y$ is acceptable.
A repeated $x$ with a different $y$ means the relation is not a function.
This scan works for both ordered-pair lists and tables.

Mapping Diagrams

A mapping diagram draws two ovals side by side. The left oval lists all domain values (inputs); the right oval lists all range values (outputs). Arrows connect each input to its output(s).

In a function, every input arrow leads to exactly one output bubble. If any input bubble has two or more arrows leaving it (pointing to different outputs), the relation is not a function.

It is acceptable for two arrows from different inputs to point to the same output bubble — that just means two inputs share an output, which is allowed.

Mapping diagrams make the function rule very visual: look at the left oval and check that no bubble has more than one arrow leaving it.

Each input bubble must have exactly one outgoing arrow for the relation to be a function.
Multiple arrows arriving at one output bubble is fine.
One input with two outgoing arrows means it is not a function.

The Vertical Line Test for Graphs

When a relation is shown as a graph on the Cartesian plane, you can use the vertical line test to decide whether it is a function. The test works because every vertical line on the plane represents a single

x

-value. If a vertical line crosses the graph in more than one point, then that one

x

-value produces more than one

y

-value — a violation of the function rule.

To apply the test, imagine sweeping a vertical line from left to right across the entire graph. If the line touches the graph in exactly one point no matter where you place it, the relation is a function. If there is even one position where the vertical line touches two or more points, the relation is not a function.

A classic example: the graph of

y = x^2

(a parabola opening upward) passes the vertical line test — every vertical line hits the parabola at most once. However, the graph of a circle such as

x^2 + y^2 = 25

fails the test because a vertical line drawn through the interior of the circle crosses it at two points (one above the

x

-axis and one below).

Remember: the vertical line test is a graphical shortcut for the same rule — one input, one output. It does not introduce a new idea; it just makes the rule easy to see on a graph.

Draw (or imagine) vertical lines sweeping across the graph.
One intersection point at every position means it is a function.
Two or more intersection points at any position means it is not a function.
The test is a visual version of the one-input, one-output rule.

Summary: How to Test Each Representation

Representation	What to Look For	Not a Function If…
Ordered pairs	Check whether any $x$ -value appears more than once with a different $y$ -value	Same $x$ , different $y$ in two pairs
Table of values	Scan the input column for repeated values with different outputs	Repeated $x$ paired with a different $y$
Mapping diagram	Count the arrows leaving each input bubble	Any input bubble has two or more outgoing arrows
Graph	Apply the vertical line test across the whole graph	Any vertical line crosses the graph at two or more points

Worked example

Example 1 — Ordered Pairs and a Mapping Diagram (Mixed Difficulty)

Two relations are given below. For each one, state whether it is a function and justify your answer using the ordered pairs. Then sketch a mapping diagram for Relation B and explain what the diagram shows.

Relation A:

\{(-3, 6),\ (0, 2),\ (4, 6),\ (7, -1)\}

Relation B:

\{(1, 5),\ (2, 8),\ (1, -3),\ (4, 0)\}

List the inputs for Relation A
Write out the $x$ -values from each ordered pair in Relation A: $-3,\ 0,\ 4,\ 7$ . Check whether any value repeats.
$x ∈ \{-3,\ 0,\ 4,\ 7\}$
Apply the function rule to Relation A
All four $x$ -values are different, so no input is paired with two different outputs. Even though $y = 6$ appears for both $x = -3$ and $x = 4$ , the rule only checks inputs, not outputs. Relation A is a function.
List the inputs for Relation B
Write out the $x$ -values from Relation B: 1,\ 2,\ 1,\ 4. Notice that $x = 1$ appears twice.
$x ∈ \{1,\ 2,\ 1,\ 4\}$
Check whether the repeated input causes a problem
The input $x = 1$ is paired with $y = 5$ in the first pair and with $y = -3$ in the third pair. Because one input maps to two different outputs, the function rule is broken. Relation B is not a function. x = 1 \Rightarrow y = 5 and y = -3
Interpret the mapping diagram for Relation B
In the mapping diagram, the input bubble labelled $1$ has two arrows leaving it — one pointing to $5$ and one pointing to $-3$ . This single input with two outgoing arrows is the visual proof that Relation B is not a function. All other input bubbles ( $2$ and $4$ ) each have only one arrow, which is fine on its own, but the double arrow from $1$ is enough to disqualify the entire relation.

Answer: Relation A is a function (all inputs are unique). Relation B is not a function because the input

x = 1

maps to both

y = 5

and

y = -3

.

Check: Re-examine each ordered pair in Relation A:

(-3, 6)

,

(0, 2)

,

(4, 6)

,

(7, -1)

. No

x

-value repeats — confirmed function. In Relation B, the pairs

(1, 5)

and

(1, -3)

share the same

x = 1

but have different

y

-values — confirmed non-function.

Worked example

Example 2 — Graphs and the Vertical Line Test (Mixed Difficulty)

Three graphs are described below. Apply the vertical line test to each and state whether it represents a function. Explain your reasoning for each case.

Graph P: a straight line with equation

y = 2x - 1

Graph Q: a circle centred at the origin with radius

4

, described by

x^2 + y^2 = 16

Graph R: a curve where

x = y^2

(a sideways parabola opening to the right)

Test Graph P with a vertical line
The equation $y = 2x - 1$ is a straight line that is not vertical. For any $x$ -value you choose, substituting it into $y = 2x - 1$ gives exactly one $y$ -value. A vertical line will cross a non-vertical straight line at exactly one point. Graph P passes the vertical line test.
$y = 2x - 1$
Conclude for Graph P
Because every vertical line meets the graph at exactly one point, Graph P is a function.
Test Graph Q with a vertical line
Pick $x = 0$ and substitute into $x^2 + y^2 = 16$ . You get $y^2 = 16$ , so $y = 4$ or $y = -4$ . Two different outputs come from one input. A vertical line at $x = 0$ crosses the circle at the point $(0, 4)$ and at $(0, -4)$ — two intersections. 0^2 + y^2 = 16 \Rightarrow y = 4 or y = -4
Conclude for Graph Q
Because there exists at least one vertical line that crosses the graph at two points, Graph Q fails the vertical line test. The circle is not a function.
Test Graph R with a vertical line
For the curve $x = y^2$ , try $x = 9$ . Then $9 = y^2$ , giving $y = 3$ or $y = -3$ . The point $(9, 3)$ and the point $(9, -3)$ are both on the curve, so a vertical line at $x = 9$ crosses the graph twice. 9 = y^2 \Rightarrow y = 3 or y = -3
Conclude for Graph R
Because the vertical line at $x = 9$ hits two points on the curve, Graph R fails the vertical line test. The sideways parabola $x = y^2$ is not a function.

Answer: Graph P (line

y = 2x - 1

) is a function. Graph Q (circle

x^2 + y^2 = 16

) is not a function. Graph R (sideways parabola

x = y^2

) is not a function.

Check: For Graph P, any chosen

x

gives exactly one

y = 2x - 1

. For Graph Q,

x = 0

gives

y = \pm 4

— two outputs. For Graph R,

x = 9

gives

y = \pm 3

— two outputs. All three conclusions are confirmed.

Common mistakes and how to avoid them

Thinking a relation is not a function because two inputs share the same output (e.g., seeing

y = 4

appear twice and calling it a non-function).

Correction: Repeated outputs are completely allowed. Only repeated inputs with different outputs violate the function rule. Always check the

x

-values (inputs), not the

y

-values (outputs).

Applying the vertical line test to only part of the graph and missing a section where two points line up vertically.

Correction: Sweep an imaginary vertical line across the entire graph from left to right. Check every part of the domain, not just the middle or the visible centre.

Confusing a horizontal line test with the vertical line test.

Correction: For identifying functions from graphs, always use a vertical line. Horizontal lines test a different property that is not part of this expectation.

Treating a mapping diagram with two inputs pointing to the same output as a non-function.

Correction: In a mapping diagram, the rule applies to outgoing arrows (from input to output), not incoming arrows. Multiple inputs pointing to one output is fine.

Assuming any curved graph is automatically not a function.

Correction: Curves can absolutely be functions. For example,

y = x^2

is a curve and a function. Always apply the vertical line test rather than judging by the shape alone.

Lesson summary

A relation is a function when every input in the domain is paired with exactly one output in the range.
To test ordered pairs or a table: check that no $x$ -value repeats with a different $y$ -value.
To test a mapping diagram: confirm that every input bubble has exactly one outgoing arrow.
To test a graph: apply the vertical line test — if any vertical line crosses the graph at two or more points, the relation is not a function.
Repeated outputs are allowed; it is repeated inputs with different outputs that break the function rule.
All four representations — ordered pairs, tables, mapping diagrams, and graphs — express the same one-input, one-output rule in different visual forms.

Check your understanding

Question 1

Which of the following sets of ordered pairs represents a function?

$\{(2, 3),\ (2, 7),\ (5, 1)\}$
$\{(1, 4),\ (3, 4),\ (5, 4)\}$
$\{(0, 6),\ (4, 9),\ (0, -2)\}$
$\{(7, 1),\ (8, 2),\ (7, 3)\}$

Show answer and explanation

\{(1, 4),\ (3, 4),\ (5, 4)\}

In option B, the inputs are CAD 1, 3, 5 — all different. Even though the output

y = 4

repeats, each input gives exactly one output, so it is a function. Options A, C, and D all have a repeated

x

-value (

2

,

0

, and

7

respectively) paired with different

y

-values, which violates the function rule.

Question 2

A mapping diagram shows the input

6

with two arrows — one pointing to

10

and one pointing to

-10

. What does this tell you?

The relation is a function because the two outputs are opposites of each other.
The relation is a function because the domain has only one value.
The relation is not a function because one input maps to two different outputs.
The relation is not a function because the range contains negative numbers.

Show answer and explanation

The relation is not a function because one input maps to two different outputs.

One input (

6

) is connected to two different outputs (

10

and

-10

). This breaks the rule that each input must map to exactly one output. The negative output and the number of domain values are irrelevant to the function test.

Question 3

You draw a vertical line at

x = 3

on a graph and it crosses the curve at the points

(3, 5)

and

(3, -5)

. What can you conclude?

The graph is a function because the two $y$ -values are equal in absolute value.
The graph is not a function because the vertical line crosses it at two points.
You need to test more vertical lines before making a conclusion.
The graph is a function because only one $x$ -value was tested.

Show answer and explanation

The graph is not a function because the vertical line crosses it at two points.

Finding even one vertical line that crosses the graph at two points is enough to conclude the relation is not a function. The input

x = 3

produces two outputs,

y = 5

and

y = -5

, which violates the one-input, one-output rule.

Question 4

The graph of

y = x^2

is a parabola opening upward. Is it a function? Why?

No, because the parabola is curved and curves are never functions.
No, because both $x = 2$ and $x = -2$ give the same output $y = 4$ .
Yes, because every vertical line crosses the parabola at exactly one point.
Yes, because the parabola crosses the $x$ -axis at the origin.

Show answer and explanation

Yes, because every vertical line crosses the parabola at exactly one point.

The vertical line test confirms that every vertical line meets the upward-opening parabola

y = x^2

at exactly one point, so it is a function. The shape being curved does not disqualify it, and having two inputs share an output (option B describes an allowed situation) is not a violation of the function rule.

Key terms

Relation: Any set of pairings between input values and output values; written as a set of ordered pairs $(x, y)$ .
Function: A relation in which every input value is paired with exactly one output value.
Domain: The set of all input values ( $x$ -values) of a relation.
Range: The set of all output values ( $y$ -values) of a relation.
Ordered pair: A pair of values written as $(x, y)$ that represents one input and its corresponding output.
Mapping diagram: A visual representation of a relation using two ovals connected by arrows, where the left oval shows inputs and the right oval shows outputs.
Vertical line test: A graphical method for identifying functions: if every vertical line crosses a graph at most once, the graph represents a function.
Non-function: A relation in which at least one input value is paired with two or more different output values.

Continue through MCR3U

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A1.1. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

A1.1 · Distinguish functions from non-functions using multiple representations

Ontario Grade 11 Mathematics

Distinguishing Functions from Non-Functions Using Multiple Representations

What you will learn

Prerequisite Bridge: Relations and Ordered Pairs

The Definition of a Function

Testing Ordered Pairs and Tables of Values

Mapping Diagrams

The Vertical Line Test for Graphs

Summary: How to Test Each Representation

Example 1 — Ordered Pairs and a Mapping Diagram (Mixed Difficulty)

Example 2 — Graphs and the Vertical Line Test (Mixed Difficulty)

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson