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

01 / LEARN

The idea, step by step

A relation need not be a function

A relation is a collection of input-output pairs. It is a function of x when each permitted x has exactly one corresponding y. Different inputs may share the same output; that does not violate the definition. What is forbidden is one input being assigned two different outputs.

For a finite table, group entries by input and check for conflicting outputs. Repeating the same pair does not cause a conflict, but the pairs (2,1) and (2,5) do. State which variable is being treated as the input; a relation may be a function of y without being a function of x.

A graphical test

A vertical line represents a fixed x-value. If it crosses the graph in more than one point, that input has more than one output, so the relation is not a function of x. A horizontal line instead tests whether outputs are repeated; it is not the test for whether the relation is a function.

A circle is not a function of x because most vertical lines through it meet an upper and a lower point. Restricting it to its upper semicircle produces a function. A table sampled from a curve is only evidence about its listed values, not proof about every unlisted input.

A common trap

Rejecting a function because two different inputs have the same output.

Instead: Only conflicting outputs for the same input break the function definition. Repeated outputs are relevant to one-to-one behaviour, not to being a function.

02 / WORKED EXAMPLES

See the thinking

Example 1

Repeated outputs are allowed

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

Each of the three inputs appears with one output.
The repeated output 4 is permitted because it comes from two different inputs.

Yes.

Example 2

A circle and a semicircle

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

At x=0, the circle permits y=3 and y=-3.
The square-root rule selects only the non-negative output, so each permitted input has one output.

The full circle is not a function of x; the upper semicircle is, on

- 3 \leq x \leq 3

Check your understanding

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

Show answer

No.

The input 1 is assigned both 2 and 3.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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)=2x+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)=2x+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 ↗

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