Test the allowed inputs.

A relation may give several outputs for one input; a function cannot. A vertical line therefore meets a function graph at most once. For y = $\sqrt{x-h}$ + k, the real domain begins at h. A full circle is a relation, but not a function of x.

A reciprocal function a/(x − h) + k excludes x = h and has asymptotes x = h and y = k. A piecewise rule uses different formulas on different parts of its domain. Check whether each endpoint is included and whether two formulas agree at a join.

INTERACTIVE MODEL

Try it. Watch it change.

y =

\sqrt{x}

RelationVertical probe x1

1 output at x = 1 · A function on its domain

The vertical line test checks how many outputs correspond to each input. A gap can indicate an excluded input; the reciprocal has an asymptote at x = 0.

Explore: Use the vertical probe on each relation. Compare excluded inputs and numbers of outputs.

Circle

(x-h)^2+(y-k)^2=r^2

Reciprocal function

y=\frac a{x-h}+k

WORKED EXAMPLE

Let $f(x)$ = x + 1 for x < 2, and $f(x)$ = $x^{2}$ for x ≥ 2. Find $f(2)$ .

The input 2 belongs to the second branch.
$f(2)$ = $2^{2}$ = 4; the first branch’s limiting value 3 is not $f(2)$ .

Assumed knowledge

Function notation, inequalities and coordinates.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use domain, range and the vertical line test.
Interpret piecewise, square-root, circle and reciprocal relations.

QCAA Mathematical Methods 2025 v1.3 · p. 17 ↗

READY TO TRY IT?

Test the allowed inputs.

Let f(x)f(x)f(x) = x + 1 for x < 2, and f(x)f(x)f(x) = x2x^{2}x2 for x ≥ 2. Find f(2)f(2)f(2).

Put the idea to work.

Let $f(x)$ = x + 1 for x < 2, and $f(x)$ = $x^{2}$ for x ≥ 2. Find $f(2)$ .