Distinguish functions from non-functions

Ontario Grade 11 Mathematics

Quadratic Functions

How to tell whether each input has exactly one output

A relation connects inputs to outputs. For example, a rule might connect the number of items bought to their total cost. A function is a special kind of relation: every allowed input is matched with exactly one output. In this lesson, you will use that idea to distinguish functions from non-functions in tables, graphs, and equations. The central question is always the same: can one input lead to more than one output?

What you will learn

Explain what makes a relation a function.
Classify a relation shown as a table, graph, or equation.
Use the vertical line test to check a graph.
Describe why a relation fails to be a function when one input has more than one output.

1. Prerequisite bridge: inputs, outputs, and ordered pairs

An input is a value you put into a relation. An output is a value connected to that input. In a table, the input is often shown in an

x

column and the output in a

y

column. In an ordered pair, the first value is the input and the second is the output.

For example, the ordered pair

(3, 8)

says that input

3

is connected to output

8

. A relation is a set of such connections. It can be shown using ordered pairs, a table, a graph, or an equation.

A function does not need to give a different output for every input. Two different inputs may share one output. The important rule is that a single input cannot be connected to two different outputs.

The first value in an ordered pair is the input; the second is the output.
Check whether any one input is paired with more than one output.

2. The rule for deciding

A relation is a function if each input has exactly one output. “Exactly one” means there is an output, and there is not a second, different output for that same input.

A relation is a non-function if at least one input is paired with two or more different outputs. One repeated input with the same output does not create a problem. It is the different outputs for one input that break the function rule.

Think of a machine that accepts an input and produces an output. For the relation to be a function, using the same input must not produce two different outputs. The machine may send different inputs to the same output.

x\mapsto y

One input must have one output.
Different inputs may have the same output.
A single input linked to two different outputs makes the relation a non-function.

3. Check tables, graphs, and equations

In a table or list of ordered pairs, compare the outputs whenever an input repeats. If the repeated input has different outputs, the relation is not a function. If no input has conflicting outputs, it is a function.

On a graph, each point represents an input-output pair. Imagine drawing vertical lines across the graph. A vertical line fixes one input value. If any vertical line crosses the graph at two or more points, that input has more than one output, so the graph is not a function. If every vertical line crosses at most once, the graph represents a function.

The phrase “at most once” matters. A vertical line does not have to cross the graph for every possible input. The test only rules out a vertical line that crosses more than once.

For an equation, try to determine whether one input value can produce two different output values. For instance, the equation

y=x^2

gives one output for each input. The equation

x=y^2

can give two outputs for some inputs: when

x=9

, both

y=3

and

y=-3

work. Therefore, the relation described by

x=y^2

is not a function of

x

.

Quadratic, exponential, and sine equations can all describe functions when each input in their domain has just one output. Their graphs may look very different, but the same input-output rule applies. A graph or a table can help you check the rule without relying only on how an equation looks.

\text{one input}\;\longrightarrow\;\text{one output}

Table or ordered pairs: look for a repeated input with different outputs.
Graph: a vertical line crossing twice shows a non-function.
Equation: test whether an input can produce more than one output.

4. Apply the rule and explain your decision

When you classify a relation, state the evidence. For a table, name the repeated input and its outputs. For a graph, describe the vertical line that crosses more than once, or state that every vertical line crosses at most once. For an equation, show an input that gives two outputs if the relation is a non-function.

A real situation can also be described by a relation. Suppose an input is a person's name and the output is the person's birth month. Each person has one birth month, so this relation follows the function rule. If the output instead records all the different activities a person enjoys, one person could have several outputs. That relation would not be a function under this input-output choice.

The choice of input and output matters when interpreting a situation. Do not decide from the topic alone. Identify what counts as an input and what counts as an output, then check whether each input has exactly one output.

Give evidence for your classification rather than relying on appearance.
In an application, identify the input and output before checking the rule.

A quick input-output check

Representation	What to check	Decision
Pairs or table	Does a repeated input have different outputs?	If yes, non-function
Graph	Does a vertical line cross more than once?	If yes, non-function
Equation	Can one input produce different outputs?	If yes, non-function

Worked example

Classifying a relation in two forms

A relation is shown by the pairs

(1, 4)

,

(2, 7)

,

(3, 4)

, and

(2, 9)

. Decide whether it is a function. Then describe what would happen if these pairs were plotted.

Compare repeated inputs
The input $2$ appears twice. Its outputs are $7$ and $9$ , which are different. That means one input is connected to more than one output.
$2\mapsto 7,\quad 2\mapsto 9$
Classify the relation
Because the input $2$ has two different outputs, the relation does not meet the function rule. The fact that input $1$ and input $3$ share output $4$ is allowed; different inputs may have the same output.
$1\mapsto 4,\quad 3\mapsto 4$
Connect the pairs to the graph
On the graph, the two points with input coordinate $2$ lie at different heights. A vertical line at $x=2$ would cross both points. This is another way to see that the relation is a non-function.
$x=2$

Answer: The relation is a non-function because input

2

has outputs

7

and

9

.

Check: The pair of points with input coordinate

2

makes the vertical line at

x=2

cross the graph twice.

Common mistakes and how to avoid them

Thinking that every output must be different.

Correction: Different inputs may share an output. The rule only requires each input to have one output.

Treating a repeated input as automatically a problem.

Correction: A repeated input is allowed if it is paired with the same output each time. It is a problem only when the outputs differ.

Using a horizontal line to test whether a graph is a function.

Correction: Use vertical lines. A vertical line checks whether one input value has more than one output.

Deciding from the shape or name of an equation alone.

Correction: Check the input-output rule, or use a graph or table to look for an input with multiple outputs.

Lesson summary

A function assigns exactly one output to each input.
A relation is a non-function if one input has two or more different outputs.
In pairs and tables, compare outputs for repeated inputs.
On a graph, a vertical line crossing more than once shows a non-function.
Always identify the inputs and outputs before classifying a relation.

Check your understanding

Question 1

Is the relation

(0, 5)

,

(1, 5)

,

(2, 8)

a function?

Yes, because no input is paired with two different outputs.
No, because two inputs share output $5$ .
No, because the outputs are not all different.
Yes, only if each output appears once.

Show answer and explanation

Yes, because no input is paired with two different outputs.

Each input appears once, so each has exactly one output. It is allowed that inputs

0

and

1

share output

5

.

Question 2

A graph has two points on the vertical line

x=4

. What does this tell you?

It is a function because both points have the same input.
It is a non-function because input $4$ has two outputs.
It is a function if the outputs are positive.
Nothing can be decided from a vertical line.

Show answer and explanation

It is a non-function because input

4

has two outputs.

Both points have input

4

, but their different heights mean they have different outputs. The relation is a non-function.

Question 3

Which statement must be true for a relation to be a function?

Every input has exactly one output.
Every output has exactly one input.
No two inputs can share an output.
The relation must be shown as an equation.

Show answer and explanation

Every input has exactly one output.

The function rule is about each input having exactly one output. Outputs may be shared, and a relation can be represented in several ways.

Key terms

Input: A value that is paired with an output in a relation.
Output: A value connected to an input in a relation.
Relation: A set of connections between inputs and outputs.
Function: A relation in which every input has exactly one output.
Non-function: A relation in which at least one input has more than one different output.
Vertical line test: A graph check: if a vertical line crosses a graph more than once, the relation is not a function.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A2.1. It is a study resource, not an official curriculum publication.

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