What a function is

A function is a rule that turns each input into exactly one output. Think of a vending machine: press B4 and you always get the same snack. If B4 sometimes gave chips and sometimes gum, it wouldn't be a function.

Put 3 in, get 7 out. We write

f (3) = 7

f (3)

is not

f

times 3. It means: the output of

f

when the input is 3.

The vertical line test

On a graph, an input is a vertical line. If any vertical line hits the graph twice, one input has two outputs, so it isn't a function.

A parabola passes: each vertical line hits it at most once.

Work through it

If $f (x) = x^{2} - 3 x$ , find $f (- 2)$ .

Replace each $x$ with $- 2$ , in parentheses. $(- 2)^{2} - 3 (- 2)$

Calculate. $4 + 6 = 10$

If $h (t) = 3 t + 2$ , what input gives an output of 14?

Set the output equal to 14. $3 t + 2 = 14$

Solve. $t = 4$

Watch out for this

Reading $f (x + 1)$ as $f (x) + 1$ .

$f (x + 1)$ changes the input. $f (x) + 1$ changes the output. They're usually different.

The short version
Each input gets exactly one output. To evaluate

f (a)

, replace every

x

with

a

Your turn

Prefer paper? Print the Functions and sequences worksheet, with an answer key.

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