Rational functions | Year 12

01 / LEARN

The idea, step by step

Factor before sketching

A rational function is a quotient of polynomials. The denominator fixes excluded inputs before cancellation. A cancelled factor can leave a hole; a surviving zero denominator with nonzero numerator creates a pole.

Solve for zeros only at allowed inputs. A zero of the numerator is not an intercept if the original denominator also excludes that x.

Use division and derivatives

Polynomial division exposes horizontal or oblique asymptotes. A horizontal asymptote describes large-input behaviour and may be crossed at finite x.

Differentiate the actual simplified rule on the original domain to find stationary points. Check derivative signs on intervals separated by both critical points and excluded inputs.

A common trap

Restoring an excluded input after cancellation.

Instead: Keep the original domain and show a hole where cancellation removes the pole.

02 / WORKED EXAMPLES

See the thinking

Example 1

A removable hole

Find the hole in $f (x) = (x^{2} - 1) / (x - 1)$ .

For x not equal to 1, simplify to x+1.
The excluded original input remains excluded.

(1, 2)

Example 2

A rational function with extrema

Find the stationary points and range of $f (x) = x / (x^{2} + 1)$ .

$f^{'} (x) = (1 - x^{2}) / {(x^{2} + 1)}^{2}$ .
The sign changes at -1 and 1 establish the extrema.

Minimum

(- 1, - 1 / 2)

, maximum

(1, 1 / 2)

; range

[- 1 / 2, 1 / 2]

Check your understanding

For $(x^{2} - 9) / (x - 3)$ , is x=3 allowed?

Show answer

No.

The original denominator is zero even though the simplified rule is x+3.

03 / WORKSHEET

Your turn

6 questions · 20 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the hole in $f (x) = (x^{2} - 1) / (x - 1)$ .
3 marks
Divide $(x^{2} + 1) / (x - 1)$ and state the oblique asymptote.
3 marks
Analyse $f (x) = (2 x + 1) / (x - 3)$ : give asymptotes and intercepts.
4 marks
Find the stationary points and range of $f (x) = x / (x^{2} + 1)$ .
4 marks
For $f (x) = (x - 2) / (x^{2} - 4)$ , state the domain, simplified rule and discontinuities.
4 marks
Can a graph cross a horizontal asymptote? Use $f (x) = x / (x^{2} + 1)$ to justify your answer.
2 marks

Read the worked solutions

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

Find the hole in $f (x) = (x^{2} - 1) / (x - 1)$ .
3 marks
For x not equal to 1, simplify to x+1.
The excluded original input remains excluded.
$(1, 2)$ .
Divide $(x^{2} + 1) / (x - 1)$ and state the oblique asymptote.
3 marks
Polynomial division gives quotient x+1 and remainder 2.
The remainder term tends to zero as the magnitude of x grows.
$x + 1 + 2 / (x - 1)$ ; asymptote $y = x + 1$ .
Analyse $f (x) = (2 x + 1) / (x - 3)$ : give asymptotes and intercepts.
4 marks
Rewrite as $2 + 7 / (x - 3)$ .
Set the numerator to zero for the x-intercept; substitute x=0 for the other.
Asymptotes $x = 3$ , $y = 2$ ; intercepts $(- 1 / 2, 0)$ and $(0, - 1 / 3)$ .
Find the stationary points and range of $f (x) = x / (x^{2} + 1)$ .
4 marks
$f^{'} (x) = (1 - x^{2}) / {(x^{2} + 1)}^{2}$ .
The sign changes at -1 and 1 establish the extrema.
Minimum $(- 1, - 1 / 2)$ , maximum $(1, 1 / 2)$ ; range $[- 1 / 2, 1 / 2]$ .
For $f (x) = (x - 2) / (x^{2} - 4)$ , state the domain, simplified rule and discontinuities.
4 marks
Factor the denominator into (x-2)(x+2).
Cancellation does not restore the excluded input 2.
Domain excludes $\pm 2$ ; $f (x) = 1 / (x + 2)$ for allowed x; pole at -2 and hole at $(2, 1 / 4)$ .
Can a graph cross a horizontal asymptote? Use $f (x) = x / (x^{2} + 1)$ to justify your answer.
2 marks
A horizontal asymptote describes end behaviour, not a forbidden y-value.
Yes; it crosses its horizontal asymptote y=0 at the origin.

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Intercepts, stationary behaviour and asymptotes

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist functions and sketching graphs. Calculus-based analysis assumes concurrent Methods differentiation; real-polynomial division detail is taught in the separate Real Polynomials chapter.

Official curriculum source ↗

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