Functions versus relations and the vertical-line test | Year 11

01 / LEARN

Understand the idea.

A relation need not be a function

A relation is a set of ordered pairs. It is a function of $x$ when every permitted input $x$ is assigned exactly one output $y$ . Several different inputs may share an output; the restriction concerns conflicting outputs for the same input.

For a finite relation, group pairs by their first coordinate. Repeated identical pairs add no conflict, but $(2, 1)$ and $(2, 5)$ do. Always identify the input variable: reversing the roles of the coordinates can change whether a relation is a function.

A graphical test

A vertical line fixes one input $x$ . Two intersection points mean that input has two outputs, so the graph fails the vertical-line test. Horizontal lines instead check repeated outputs and are relevant to whether a function is one-to-one.

An entire circle fails because an interior input can have an upper and a lower output. Selecting its upper semicircle removes that ambiguity. A finite sample of a curve cannot prove the test holds at all other inputs; use the rule or full graph.

The relation x squared plus y squared equals 9 on equal axes from -5 to 5. The upper semicircle is sage and the lower semicircle is gold, joining at x=-3 and x=3.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Repeated outputs are allowed

Does $(- 1, 4), (0, 4), (2, 7)$ define a function of x?

Worked solution

1
List each distinct input and its assigned output.
$\begin{aligned} - 1 & \mapsto 4 \\ 0 & \mapsto 4 \\ 2 & \mapsto 7 \end{aligned}$
These are all pairs in the supplied finite relation.
2
Check for conflicting outputs.
Each input appears with exactly one output, so the relation is a function of its first coordinate.
3
Interpret the repeated output.
$f (- 1) = f (0) = 4$
Two different inputs sharing four is allowed; it only shows this function is not one-to-one.

Answer

Yes.

EXAMPLE 02

A circle and a semicircle

Compare $x^{2} + y^{2} = 9$ with $y = \sqrt{9 - x^{2}}$ .

Worked solution

1
Test the full circle at a single input.
$\begin{aligned} 0^{2} + y^{2} & = 9 \\ y & = \pm 3 \end{aligned}$
The input zero has two outputs, which is enough to show the full circle is not a function of x.
2
Find the square-root rule’s permitted inputs.
$\begin{aligned} 9 - x^{2} & \geq 0 \\ x^{2} & \leq 9 \\ - 3 & \leq x \leq 3 \end{aligned}$
The radicand must be non-negative for the output to be real.
3
Identify the selected branch.
$y = \sqrt{9 - x^{2}} \geq 0$
The principal square root gives exactly one non-negative output at every permitted input, so the upper semicircle is a function.

Answer

The full circle is not a function of x; the upper semicircle is, on $- 3 \leq x \leq 3$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Is $(1, 2), (1, 3), (2, 4)$ a function of its first coordinate?

Show answer & reasoning

No.

The pairs $(1, 2)$ and $(1, 3)$ assign different outputs to the same first coordinate $1$ . Therefore the given relation is not a function of its first coordinate.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Decide whether ${(0, 1), (1, 1), (2, 1)}$ is a function of $x$ .
2 marks
Decide whether ${(- 2, 0), (- 2, 4), (3, 5)}$ is a function of $x$ .
2 marks
Is the relation $x = 3$ a function of $x$ ? Explain.
2 marks
Is $y = x^{2}$ a function of $x$ , and is each output produced by only one input?
2 marks
Restrict $x^{2} + y^{2} = 16$ to give a function with non-positive outputs.
2 marks
A table lists $f (0) = 1$ , $f (1) = 3$ and $f (2) = 5$ . Does it prove $f (x) = 2 x + 1$ for every real $x$ ?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Decide whether ${(0, 1), (1, 1), (2, 1)}$ is a function of $x$ .
2 marks
Every input has exactly one output, although all outputs coincide.
Yes.
Decide whether ${(- 2, 0), (- 2, 4), (3, 5)}$ is a function of $x$ .
2 marks
Input -2 has two different outputs.
No.
Is the relation $x = 3$ a function of $x$ ? Explain.
2 marks
The same x-value 3 permits every real y-value, so a vertical line overlaps it at many points.
No.
Is $y = x^{2}$ a function of $x$ , and is each output produced by only one input?
2 marks
Every $x$ has one square.
For example, inputs 2 and -2 both produce 4.
It is a function of $x$ , but it is not one-to-one on all real inputs.
Restrict $x^{2} + y^{2} = 16$ to give a function with non-positive outputs.
2 marks
Select the lower semicircle, which assigns one non-positive y-value to each allowed $x$ .
$y = - \sqrt{16 - x^{2}}$ , $- 4 \leq x \leq 4$ .
A table lists $f (0) = 1$ , $f (1) = 3$ and $f (2) = 5$ . Does it prove $f (x) = 2 x + 1$ for every real $x$ ?
2 marks
It agrees with that rule at the listed inputs, but many other rules have those same three values.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Topic 1: Functions and graphs. Prerequisite algebra and coordinate geometry are embedded as supporting lessons.

Official curriculum source ↗

In this lesson

Understand the idea.

A relation need not be a function

A graphical test

See the method in action.

Repeated outputs are allowed

List each distinct input and its assigned output.

Check for conflicting outputs.

Interpret the repeated output.

A circle and a semicircle

Test the full circle at a single input.

Find the square-root rule’s permitted inputs.

Identify the selected branch.

Try it yourself.

Check your understanding

Keep practising