Function notation: evaluate and read f(x)

Discover

$f(x) = x + 5$ is a rule: it takes a number and adds 5.

What does $f(2)$ mean?

The output of $f$ when the input is 2
$f$ times 2
The input that gives the output 2

Key idea

A function gives exactly one output for each input. We write the output as $f(x)$ .

f(a)

is the output of

f

for the input

a

. To find it, replace every

x

in the rule by

(a)

, with parentheses.

On a graph, $f(a)$ is the height of the graph above $x = a$ .

Worked example

$f(x) = 3x^2 - x$ . Find $f(-2)$ .

f(-2) = 3(-2)^2 - (-2)

Replace each x by (−2), with the parentheses.

= 3 \cdot 4 + 2

(−2)² = 4, and subtracting −2 adds 2.

= 14

Multiply, then add.

Your turn, guided

Your turn: $f(x) = 2x^2 + 1$ . Fill in the missing line.

f(-3) = 2(-3)^2 + 1

Replace x by (−3).

f(-3) = 19

(−3)² = 9, then 2 · 9 + 1.

Practice ★

$f(x) = 2x^2 + 3$ . Find $f(-3)$ .
$f(x) = 4x^2 + 8$ . Find $f(-2)$ .
$f(x) = 4x^2 + 2$ . Find $f(-5)$ .
$f(x) = 5x^2 + 5$ . Find $f(-4)$ .
$f(x) = 3x^2 + 3$ . Find $f(-4)$ .

Watch out!

Practice ★★

$f(x) = x^2 + 3$ . Find $f(2 + 3)$ .
$f(x) = x^2 + 2$ . Find $f(4 + 4)$ .
$f(x) = x^2 + 7$ . Find $f(4 + 1)$ .
$f(x) = x^2 + 3$ . Find $f(3 + 3)$ .

Here is the graph of $y = f(x)$ . Tap the point of the graph that shows $f(2)$ .
Here is the graph of $y = f(x)$ . Tap the point of the graph that shows $f(2)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = x^2 + 3$ . Find $f(2 + 3)$ .
$f(x) = x^2 + 2$ . Find $f(4 + 4)$ .
$f(x) = x^2 + 7$ . Find $f(4 + 1)$ .

Summary card

Function notation

$f(a)$ is the output for the input $a$ .
Replace every $x$ by $(a)$ , with parentheses: $f(-2)$ for $x^2$ is $(-2)^2 = 4$ .
On a graph: the height above $x = a$ .

Watch out: $f(a + b)$ is not $f(a) + f(b)$ .

End of the lesson

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