Function

A function $f$ from a set $A$ to a set $B$, written $f\colon A \to B$, pairs every element $x$ of $A$ with exactly one element of $B$, written $f(x)$. It is also called a map or mapping. $A$ is the domain, the set of inputs; $B$ is the codomain, where the outputs may come from; the outputs actually reached form the range, or image. Here you draw the pairing as arrows from $A$ to $B$ and see the same pairs as points $(x, y)$ on a grid; the readout says whether they make a function and, if not, which input breaks the condition.

The condition is about the inputs only. Each input needs an output, and only one: an input with no arrow, or with two or more, means the pairs are not a function from $A$ to $B$. The outputs have no such condition. Several inputs may share an output, as $-2$ and $2$ share $4$ under $x^2$, and elements of $B$ may receive no arrow at all; a constant function sends every input to the same output.

Nor does a function need a formula: giving each input any one output defines a function, whether or not some formula describes it. And an equation does not guarantee one. On these sets $y^2 = x$ pairs $1$ with both $-1$ and $1$, and gives $-2$, $-1$ and $2$ nothing in $B$.

Whether a set of pairs is a function depends on $A$. Start with the function in the first highlight and remove its arrow $0 \to 4$. What is left is no longer a function from $A$ to $B$, because $0$ is in $A$ and has no output; yet the same arrows are a function on the smaller set $\{-2, -1, 1, 2\}$. Shrinking the set of inputs only settles the inputs you leave out, though: on $\{-2, -1, 1, 2\}$, $y^2 = x$ is still not a function, since $1$ still has two outputs and $-2$, $-1$ and $2$ still have none. When a function is given only by a formula, its domain is usually taken to be every $x$ for which the formula makes sense. Some areas, such as computability theory, also speak of partial functions, which may leave inputs without an output; a function from $A$ to $B$, as on this page, leaves none.

On the grid, the vertical line through an input $x$ meets one point for each output of $x$. So the pairs are a function from $A$ to $B$ exactly when, for each $x$ in $A$, the vertical line through $x$ meets exactly one point; lines through numbers outside $A$ have nothing to check. The test is the definition read off the graph, since the arrows and the points are the same pairs.