Glossary

Function

Also written: functions · f(x) · f of x

Definition

A function is a rule that assigns exactly one output to each input. If a single input could give two different outputs, the rule is not a function.

The rule that makes it a function

Each input gets exactly one output. That is the whole requirement, and the word “exactly” is doing the work: not zero outputs, and not two.

Relationship	A function?	Why
$y = 2x + 1$	yes	each $x$ gives one $y$
a student’s height by name	yes	one height per person
$x = y^2$	no	$x = 9$ gives both $y = 3$ and $y = -3$
a vertical line	no	one $x$ has infinitely many $y$

The vertical line test

Draw a vertical line anywhere on the graph. If it ever crosses the curve twice, that $x$ -value has two outputs and the relationship is not a function.

This is why a vertical line is itself not a function, even though its graph is perfectly straight — see linear vs nonlinear.

Notation

$f(x)$ names the rule and shows the input at once. It is not multiplication: $f$ has no value to multiply by, because it names a process rather than a quantity. See function notation.

A rule that could return two answers would make prediction impossible — feeding in $3$ would leave you asking “which output?”. Insisting on one keeps every later technique, from graphing to calculus, well defined.

Lessons that use this term

Function Notation: What f(x) Actually Means
Understand f(x) as a name and an input rather than multiplication, how to evaluate it, and how to read statements like f(3) = 7 in context.
Domain and Range: The Inputs and Outputs of a Function
What domain and range mean, how to find them from a graph, a table or an equation, and why some inputs have to be excluded.
Linear vs Nonlinear Functions: How to Tell Them Apart
Four reliable tests for deciding whether a relationship is linear — from its equation, its table, its graph, or its rate of change.