One input. One output.

A function assigns exactly one output to each input in its domain. $f(3)$ means the output at input 3; it is not f multiplied by 3. The range consists of the outputs actually obtained.

Restrictions can come from algebra or from context. Division by zero is undefined; a real square root requires a non-negative radicand. A physical model might also restrict time to t ≥ 0. This vocabulary prepares you for senior Methods.

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: Compare a circle and a square-root graph with the vertical probe. Count outputs for one x-value.

A function maps an input to an output

x\;\xrightarrow{\ f\ }\;f(x)

WORKED EXAMPLE

For $f(x)$ = $\sqrt{x-2}$ , give the real domain and find $f(11)$ .

x − 2 ≥ 0, so x ≥ 2.
$f(11)$ = $\sqrt{9}$ = 3.

Assumed knowledge

Substitution, coordinates and square roots.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use function notation and restrictions.
Connect rules, tables and graphs.

Year 10 preparation ↗

READY TO TRY IT?

One input. One output.

For f(x)f(x)f(x) = x−2\sqrt{x-2}x−2​, give the real domain and find f(11)f(11)f(11).

Put the idea to work.

For $f(x)$ = $\sqrt{x-2}$ , give the real domain and find $f(11)$ .