Domain and range | Year 11

01 / LEARN

The idea, step by step

Allowed inputs come first

The domain is the set of inputs that the rule allows. For real-valued functions, a denominator cannot be zero and a quantity inside an even square root must be non-negative. A contextual restriction can be stronger than an algebraic one: a side length must normally be positive even if a formula can be evaluated at a negative number.

Write restrictions before simplifying. For $(x^{2} - 4) / (x - 2)$ , cancellation produces x+2 only when x is not 2. The excluded input does not become allowed merely because a factor has disappeared from the simplified expression.

Find what the outputs can do

The range is the set of outputs actually produced by permitted inputs. A square is non-negative, so ${(x - 3)}^{2} + 1$ has smallest output 1 when all real inputs are allowed. On a restricted domain, that minimum might no longer be reached, and an endpoint may create a largest value.

Check whether a boundary value is included. A closed interval includes its endpoints; a strict inequality excludes them. When using a graph, inspect open or closed endpoints and turning points rather than reading only the values at the edges of the picture.

A common trap

Using the unrestricted quadratic range after the input interval has changed.

Instead: Recheck the turning point and endpoint values on the stated domain. A turning point outside the interval cannot determine its minimum.

02 / WORKED EXAMPLES

See the thinking

Example 1

A denominator restriction

State the domain and range of $f (x) = 6 / (x - 2)$ .

The denominator is zero at x=2, which is excluded.
A non-zero numerator divided by a finite non-zero number cannot give zero. Conversely, any non-zero y gives the permitted input $x = 2 + 6 / y$ .

Domain: all real x except 2. Range: all real outputs except 0.

Example 2

A restricted quadratic

Find the domain and range of $g (x) = {(x - 1)}^{2} + 2$ for $- 1 \leq x \leq 3$ .

The minimum occurs at x=1 and is 2.
Both endpoints give 6. The curve is continuous, so all outputs from 2 to 6 are produced.

Domain

[- 1, 3]

; range

[2, 6]

Check your understanding

Find the real domain of $h (x) = \sqrt{x + 4}$ .

Show answer

$x \geq - 4$ .

Require $x + 4 \geq 0$ so the square root is real.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

State the real domain of $f (x) = 5 / (x + 3)$ .
2 marks
State the real domain of $g (x) = \sqrt{7 - 2 x}$ .
2 marks
Find the range of $p (x) = - {(x + 2)}^{2} + 9$ for all real x.
2 marks
Find the range of $q (x) = 3 x - 1$ on $2 \leq x \leq 5$ .
2 marks
Simplify $(x^{2} - 9) / (x - 3)$ and retain its original domain restriction.
2 marks
A square has perimeter P=4s. State a sensible domain and range when side length s is measured in centimetres.
2 marks

Read the worked solutions

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

State the real domain of $f (x) = 5 / (x + 3)$ .
2 marks
Exclude the input that makes x+3 zero.
$x \in ℝ$ , $x \neq - 3$ .
State the real domain of $g (x) = \sqrt{7 - 2 x}$ .
2 marks
Require $7 - 2 x \geq 0$ , then solve the inequality.
$x \leq 7 / 2$ .
Find the range of $p (x) = - {(x + 2)}^{2} + 9$ for all real x.
2 marks
A square is non-negative; its negative is non-positive. Equality occurs at x=-2.
$p (x) \leq 9$ .
Find the range of $q (x) = 3 x - 1$ on $2 \leq x \leq 5$ .
2 marks
The function increases; $q (2) = 5$ and $q (5) = 14$ .
$5 \leq q (x) \leq 14$ .
Simplify $(x^{2} - 9) / (x - 3)$ and retain its original domain restriction.
2 marks
Factor the numerator as (x-3)(x+3).
Cancellation is valid only when x-3 is non-zero.
$x + 3$ , with $x \neq 3$ .
A square has perimeter P=4s. State a sensible domain and range when side length s is measured in centimetres.
2 marks
A non-degenerate square has positive side length, and multiplying by 4 preserves positivity.
$s > 0$ and $P > 0$ .

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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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