What makes something a function?

Each input must have exactly one output. A rule that could give two different answers for the same input is not a function.

It is the output of the function f for the input x. If f(x) = x^2, then f(3) = 9.

Graphs & Functions

What Is a Function? Inputs, Outputs, f(x) and the Vertical Line Test

Functions as machines, how to read and evaluate $f(x)$ notation, the vertical line test for graphs, and piecewise functions that use different rules on different inputs.

3 min read Supports 4 lessons

Think of a function as a machine. You feed in an input, the machine applies its rule, and out comes exactly one output. "Double it and add 1" is a function: $3$ goes in, $7$ comes out; $-2$ goes in, $-3$ comes out.

x \;\longmapsto\; \boxed{\;2x + 1\;} \;\longmapsto\; f(x)

Key idea

The defining rule: each input has one output. Two inputs may share an output ( $x^2$ sends both $2$ and $-2$ to $4$ ), but one input may never produce two outputs.

Function notation

$f(x) = 2x + 1$ names the function $f$ and states its rule. $f(3)$ is read " $f$ of $3$ " and means the output when the input is $3$ .

Watch out

$f(x)$ does not mean $f$ times $x$ . The brackets hold the input.

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

1/4

You can also work backwards. If $g(x) = 2x + 1$ and $g(x) = 9$ , then $2x + 1 = 9$ , so $x = 4$ . And you can feed in expressions: $g(a + 1) = 2(a + 1) + 1 = 2a + 3$ .

The vertical line test

On a graph, an input is an $x$ -value and its output is the height above it. If a vertical line crosses the graph twice, that input has two outputs, so the graph is not a function.

At $x = 2$ the sideways parabola has two heights, so it fails the test.

Key idea

A graph is a function if every vertical line crosses it at most once.

Piecewise functions

A piecewise function uses different rules on different parts of its domain. Taxi fares, tax bands and phone tariffs often work like this.

f(x) = \begin{cases} x + 2 & x < 1 \\ 4 - x & x \ge 1 \end{cases}

The two pieces meet at $(1, 3)$ , so there is no jump.

Worked example

Using $f$ above, find $f(-2)$ and $f(3)$ .

1
$-2 < 1$ , so use the first piece: $f(-2) = -2 + 2 = 0$ .

The absolute value function is piecewise in disguise: $|x|$ is $x$ when $x \ge 0$ and $-x$ when $x < 0$ .

Tip

At a boundary, check whether the pieces meet. If they do, the graph joins up there, which is what calculus calls being continuous.

Common questions

What makes something a function?: Each input must have exactly one output. A rule that could give two different answers for the same input is not a function.
What does f(x) mean?: It is the output of the function $f$ for the input $x$ . If $f(x) = x^2$ , then $f(3) = 9$ .

Practise functions

The What Is a Function? lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Graphs & Functions

What Is a Function? Inputs, Outputs, f(x) and the Vertical Line Test

Functions as machines, how to read and evaluate $f(x)$ notation, the vertical line test for graphs, and piecewise functions that use different rules on different inputs.

3 min read Supports 4 lessons

x \;\longmapsto\; \boxed{\;2x + 1\;} \;\longmapsto\; f(x)

Key idea

The defining rule: each input has one output. Two inputs may share an output ( $x^2$ sends both $2$ and $-2$ to $4$ ), but one input may never produce two outputs.

Function notation

$f(x) = 2x + 1$ names the function $f$ and states its rule. $f(3)$ is read " $f$ of $3$ " and means the output when the input is $3$ .

Watch out

$f(x)$ does not mean $f$ times $x$ . The brackets hold the input.

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

1/4

You can also work backwards. If $g(x) = 2x + 1$ and $g(x) = 9$ , then $2x + 1 = 9$ , so $x = 4$ . And you can feed in expressions: $g(a + 1) = 2(a + 1) + 1 = 2a + 3$ .

The vertical line test

On a graph, an input is an $x$ -value and its output is the height above it. If a vertical line crosses the graph twice, that input has two outputs, so the graph is not a function.

At $x = 2$ the sideways parabola has two heights, so it fails the test.

Key idea

A graph is a function if every vertical line crosses it at most once.

Piecewise functions

A piecewise function uses different rules on different parts of its domain. Taxi fares, tax bands and phone tariffs often work like this.

f(x) = \begin{cases} x + 2 & x < 1 \\ 4 - x & x \ge 1 \end{cases}

The two pieces meet at $(1, 3)$ , so there is no jump.

Worked example

Using $f$ above, find $f(-2)$ and $f(3)$ .

1
$-2 < 1$ , so use the first piece: $f(-2) = -2 + 2 = 0$ .

The absolute value function is piecewise in disguise: $|x|$ is $x$ when $x \ge 0$ and $-x$ when $x < 0$ .

Tip

At a boundary, check whether the pieces meet. If they do, the graph joins up there, which is what calculus calls being continuous.

Common questions

What makes something a function?: Each input must have exactly one output. A rule that could give two different answers for the same input is not a function.
What does f(x) mean?: It is the output of the function $f$ for the input $x$ . If $f(x) = x^2$ , then $f(3) = 9$ .

Practise functions

The What Is a Function? lesson has worked examples and unlimited practice that adapts to you.

Open the lesson