What is a function

A function from a set $A$ to a set $B$ is a rule that assigns to each element of $A$ exactly one element of $B$. We write $f: A \to B$ and call $A$ the domain and $B$ the codomain.

The word "rule" here is deliberately broad - a function does not have to be defined by an equation. A table, a graph, or a verbal description can all define functions, as long as each input is assigned exactly one output. Any such rule is called a mapping, and the terms "function" and "mapping" are used interchangeably.

This is the only requirement. In particular:

• The same input cannot produce different outputs. If a rule assigns both $2$ and $-2$ to the same input, it is not a function.
• Different inputs may share the same output. The definition requires each input to have exactly one output, but does not require each output to come from exactly one input.
• Every input must have an output. A function cannot leave an input unassigned.

Does the rule $f(x) = x^2$ for $x \in \mathbb{R}$ define a function?

Solution

For any $x \in \mathbb{R}$, squaring gives exactly one real number. Each input is assigned a single output, so $f$ is a function.

Different inputs can share the same output: $f(-3) = f(3) = 9$. This is allowed - the definition requires each input to have one output, not that each output comes from one input.

When a function is defined by a formula, its graph is the set of all input-output pairs $(x, f(x))$ plotted as points in the plane. We can use this to determine whether a graph represents a function without knowing the formula - by checking whether each $x$-value corresponds to at most one $y$-value.

Use the visualisation below to see this in action. Drag the red diamond on the $x$-axis across the plane and use the buttons to cycle through different curves.

The following points are plotted on a coordinate plane. Determine whether each set represents a function.

Set A: $\{(1, 3),\; (2, 5),\; (3, 2),\; (4, 5)\}$

Set B: $\{(1, 3),\; (2, 5),\; (2, -1),\; (4, 5)\}$

Solution

Set A

We apply the vertical line test by checking whether any $x$-coordinate appears more than once.

The $x$-coordinates are $1$, $2$, $3$, $4$ - all distinct. A vertical line at each of these values crosses exactly one point.

Every vertical line passes through at most one point. Set A is a function.

Set B

The $x$-coordinates are $1$, $2$, $2$, $4$. The value $x = 2$ appears twice, with different $y$-values: $(2, 5)$ and $(2, -1)$.

The vertical line at $x = 2$ crosses two points. The input $2$ has two different outputs ($5$ and $-1$), so the vertical line test fails.

Set B is not a function.