Function Notation

Function notation gives a compact way to name a rule and ask about its output. If $f(x)=2x+1$ , then $f(3)$ means “the output of f when the input is 3.”

The parentheses identify the input. They do not mean multiplication. The same notation also works with variable or algebraic inputs such as $f(a+1)$ .

On this page (8 sections)

Evaluating a function

For the rule $f(x)=2x+1$ , the notation $f(3)$ asks for its output at input $3$ .

f(3)=2(3)+1=7

Replace every occurrence of the input symbol. With the separate rule $g(x)=x^2$ , a negative input gives $g(-3)=(-3)^2=9$ , not $-9$ . The parentheses preserve the entire input.

The symbols $f$ and $f(x)$ play different roles: $f$ names the function, while $f(x)$ is an output. The input letter can change without changing the rule: $f(t)=2t+1$ describes the same assignment.

Inputs can be expressions

If the input is $a+1$ , substitute the complete expression wherever $x$ appears.

f(a+1)=2(a+1)+1=2a+3

Adding to the input and adding to the output

f(a)+1=(2a+1)+1=2a+2

These are different expressions. The location of the addition matters: $f(a+1)$ increases the input first; $f(a)+1$ increases the output afterwards.

g(a+1)=(a+1)^2=a^2+2a+1\qquad\text{for }g(x)=x^2

When a function has a power, square the entire substituted expression. Do not apply the rule only to its first term.

Solving for an input

Evaluating $f(3)$ starts with a known input. Solving $f(x)=9$ starts with a known output and asks for an input.

2x+1=9\quad\Longleftrightarrow\quad2x=8\quad\Longleftrightarrow\quad x=4

Check: $f(4)=2(4)+1=9$ . The answer is an input, not a new formula for $f$ .

For a different rule, an output can have more than one input. If $g(x)=x^2$ on the real numbers, solving $g(x)=9$ gives $x=3$ or $x=-3$ .

More than one function

Different function names keep rules separate. With $f(x)=2x+1$ and $g(x)=x^2$ , $f(2)=5$ while $g(2)=4$ . The input is the same, but the rules differ.

Function notation separates the input from the rule

input
3

×2, then +1

output
7

The function name identifies the rule; the value inside parentheses identifies the input.

Diagram explained

For f(x)=2x+1, input 3 passes through the rule and produces output 7.

Reading notation from a table

A function h specified on three inputs
Input x	Output h(x)
$-1$	$4$
$0$	$2$
$2$	$4$

This table gives $h(0)=2$ . Solving $h(x)=4$ on this stated domain gives $x=-1$ or $x=2$ . No formula is needed to answer either question, and the table does not define $h(1)$ .

Common notation errors

$f(x)$ is one output, not the product $f\cdot x$ .
$f(3)$ does not mean replace every 3 with x; it means replace x with 3.
When the input is an expression, use parentheses around the whole substitution before simplifying.

Practice questions

If $f(x)=2x+1$ , find $f(3)$ .
Find $f(a+1)$ for $f(x)=2x+1$ .
If $f(x)=2x+1$ and $f(x)=9$ , find x.

Check your answers

$f(3)=2(3)+1=7$ .
$f(a+1)=2(a+1)+1=2a+3$ .
$2x+1=9$ gives $x=4$ , and $f(4)=9$ checks it.

Sources and further reading

OpenStax: Function notation. Definitions, conventions and related examples.
OpenStax Precalculus 2e. Functions, transformations, inverse functions and function families.
Wolfram Language: FunctionDomain. Domain conventions and symbolic domain checks.
Wolfram Language: FunctionRange. Range conventions and symbolic range checks.