Math Core

Writing " $y = 3x - 5$ " works, but it gets clumsy when you have several functions or want to say "the output when the input is $4$ ." Function notation gives each function a name and a compact way to write its inputs and outputs.

Naming a function

Definition

Function notation

The notation $f(x)$ , read " $f$ of $x$ ," means the output of the function $f$ when the input is $x$ . The letter $f$ is the name of the function, and $x$ is the input.

So instead of $y = 3x - 5$ , you can write

f(x) = 3x - 5.

This says the same thing: to get the output, multiply the input by $3$ and subtract $5$ . The graph of $f$ is the same as the graph of $y = 3x - 5$ , because $y$ and $f(x)$ both name the output.

Functions don't have to be called $f$ . You'll see $g(x)$ , $h(x)$ , and names that fit a situation, like $C(n)$ for a cost that depends on a number of items, or $d(t)$ for a distance that depends on time.

Common mistake

$f(x)$ does not mean $f$ times $x$ . The parentheses here hold the input; they are not multiplication. In the same way, $f(4)$ is "the output when the input is $4$ ," not " $f$ times $4$ ."

Evaluating a function

To find $f(4)$ , replace every $x$ in the rule with $4$ , then simplify. This is just substitution from Unit 1, written in new notation.

Input and output

$f(a)$ asks: the input is $a$ , what is the output? Substitute and simplify.
$f(x) = b$ asks: the output is $b$ , what is the input? Set the rule equal to $b$ and solve.

Worked example: Evaluating at numbers

Let $f(x) = 3x - 5$ . Find $f(4)$ , $f(-2)$ and $f(0)$ .

\begin{aligned} f(4) &= 3(4) - 5 = 12 - 5 = 7 \\ f(-2) &= 3(-2) - 5 = -6 - 5 = -11 \\ f(0) &= 3(0) - 5 = 0 - 5 = -5 \end{aligned}

Each result is an ordered pair on the graph of $f$ : $(4, 7)$ , $(-2, -11)$ and $(0, -5)$ . The statement $f(4) = 7$ says exactly the same thing as "the point $(4, 7)$ is on the graph."

When a rule has several $x$ 's, every one of them gets the input. Use parentheses, especially for negative inputs.

Worked example: A rule with a square

Let $g(x) = x^2 - 2x + 3$ . Find $g(-3)$ .

\begin{aligned} g(-3) &= (-3)^2 - 2(-3) + 3 && \text{substitute in every spot} \\ &= 9 + 6 + 3 && (-3)^2 = 9, \; -2(-3) = 6 \\ &= 18 \end{aligned}

Working backward: when is the output a given value?

Sometimes you know the output and need the input. Since you've already learned to solve equations, this is one more step.

Worked example: Solving f(x) = b

Let $f(x) = 3x - 5$ . For what input is $f(x) = 13$ ?

\begin{aligned} 3x - 5 &= 13 \\ 3x &= 18 \\ x &= 6 \end{aligned}

Check: $f(6) = 3(6) - 5 = 13$ . So $f(6) = 13$ , and the point $(6, 13)$ is on the graph.

Reading function values from a graph

On a graph of $y = f(x)$ , the value $f(a)$ is the height of the graph above (or below) $x = a$ . To find $f(a)$ , go to $a$ on the $x$ -axis, move straight up or down to the graph, and read the $y$ -value.

The graph of y = f(x). Reading heights: f(3) = 0, f(−2) = 1 and f(1) = −2.Open in grapher →

To answer " $f(x) = 1$ " from a graph, go the other way: find the height $1$ on the $y$ -axis, move across to the graph, and read every $x$ -value where the graph has that height. For the graph above, the horizontal line $y = 1$ meets the graph at $x = -2$ and $x = 4$ , so $f(x) = 1$ has two solutions. That's allowed: two inputs can share an output.

Functions in context

Function notation shines in real situations, because the statement carries the meaning of both numbers.

Worked example: Interpreting function notation

A bike rental shop charges a $6 helmet fee plus $14 per hour. The cost in dollars for $h$ hours is $C(h) = 14h + 6$ .

Find $C(3)$ and explain what it means.
Solve $C(h) = 76$ and explain what it means.

Solutions.

$C(3) = 14(3) + 6 = 42 + 6 = 48$ . Renting a bike for 3 hours costs $48.
$14h + 6 = 76$ , so $14h = 70$ and $h = 5$ . For $76 you can rent a bike for 5 hours.

In $C(3) = 48$ , the number inside the parentheses (hours) is the input and the number on the right (dollars) is the output. Keeping track of which is which is the whole point.

Inputs that are expressions

The input doesn't have to be a number. If $f(x) = 3x - 5$ , then $f(a + 2)$ means "replace each $x$ with $a + 2$ ":

f(a + 2) = 3(a + 2) - 5 = 3a + 6 - 5 = 3a + 1.

The parentheses around the input are essential; the distributive property does the rest.

Tip

Check an expression input with a number. If $a = 1$ , then $f(a + 2) = f(3) = 3(3) - 5 = 4$ , and $3a + 1 = 3(1) + 1 = 4$ . They match.