Function Notation

Function notation is the way math labels a rule. Instead of writing $y = 3x + 2$ , we write $f(x) = 3x + 2$ . Nothing about the rule changed — $f(x)$ is just another name for $y$ , the output. The letter $f$ names the function, and the $x$ in parentheses is the input you feed into it.

The payoff is that $f(4)$ says two things at once: use the rule named $f$ , and use $4$ as the input. That one small piece of notation lets a problem talk about specific inputs and outputs without any extra words, and it shows up in every math course from here on.

Reading the notation

In $f(x) = 3x + 2$ , the $f$ is the name of the function, $x$ is the input, and $f(x)$ is the output. Read $f(4)$ out loud as " $f$ of $4$ " — it means the output of $f$ when the input is $4$ . It does not mean $f$ times $4$ ; the parentheses here are not multiplication.

Different letters just mean different functions. A problem can use $f(x)$ , $g(x)$ , and $h(x)$ in the same breath — three names, three separate rules.

The diagram below shows a function as a machine: each input on the left is sent by the rule to exactly one output on the right.

Evaluating a function

To evaluate $f(4)$ , substitute $4$ everywhere you see $x$ , then simplify. If $f(x) = 3x + 2$ , then $f(4) = 3(4) + 2 = 14$ . Write the substitution in parentheses, especially with negative inputs — $f(-2) = 3(-2) + 2$ keeps the signs honest.

A table or a graph can define a function too. If a table pairs $x = 2$ with $f(x) = 5$ , then $f(2) = 5$ — find the input in the $x$ column and read across to the output. The table below does exactly that for the rule $f(x) = 2x + 1$ .

f(x) = 2x + 1

$x$	$f(x)$
$0$	$1$
$1$	$3$
$2$	$5$
$3$	$7$

Working backward

Some questions hand you the output and ask for the input: if $f(x) = x + 8$ , for what $x$ is $f(x) = 20$ ? Set the rule equal to the output and solve the equation: $x + 8 = 20$ , so $x = 12$ . Evaluating plugs in an input; working backward solves for one.

Worked examples

Example 1: evaluate a linear function

If $f(x) = 2x + 5$ , find $f(3)$ .

Start with the rule

f(x) = 2x + 5

Substitute

3

for

x

f(3) = 2(3) + 5

Multiply

f(3) = 6 + 5

Add

f(3) = 11

Answer: $f(3) = 11$

Example 2: a negative input

If $g(x) = x^{2} - 4$ , find $g(-3)$ .

Start with the rule

g(x) = x^{2} - 4

Substitute

-3

for

x

, in parentheses

g(-3) = (-3)^{2} - 4

Square first — a negative squared is positive

g(-3) = 9 - 4

Subtract

g(-3) = 5

Answer: $g(-3) = 5$

Example 3: work backward from an output

If $h(x) = 4x - 6$ , what value of $x$ makes $h(x) = 18$ ?

Set the rule equal to the output

4x - 6 = 18

Add

6

to both sides

4x = 24

Divide both sides by

4

x = 6

Check:

h(6) = 4(6) - 6 = 18

✓

Answer: $x = 6$

Try one yourself

Common questions

Does f(x) mean f times x?

No. In function notation the parentheses mean "input," not multiplication. $f(x)$ is read " $f$ of $x$ " and stands for the output of the function $f$ at the input $x$ .

Is f(x) the same as y?

Yes. $f(x)$ and $y$ both name the output. $y = 3x + 2$ and $f(x) = 3x + 2$ describe the same rule — function notation just adds a name, so you can tell $f$ apart from a second function $g$ in the same problem.

What's the difference between f(4) and f(x) = 4?

$f(4)$ asks you to plug in the input $4$ and compute the output. $f(x) = 4$ tells you the output is $4$ and asks which input produced it — you set the rule equal to $4$ and solve for $x$ .

Why do problems use letters like g and h instead of f?

Each letter names a different function. If a problem compares two rules, it needs two names — $f$ and $g$ — so it can say things like $f(2)$ and $g(2)$ without confusion.

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Function Notation

Reading the notation

Evaluating a function

Working backward

Worked examples

Example 1: evaluate a linear function

Example 2: a negative input

Example 3: work backward from an output

Try one yourself

Common questions

Keep learning

Graphing Functions

Domain and Range of Continuous Functions

Intercepts of Graphs

Sketching and Comparing Functions