Introduction to Functions

One output per input

Think of a function as a machine: input goes in, output comes out. The rule for being a function is strict about inputs but relaxed about outputs. An input may never produce two different outputs — but two different inputs are allowed to share the same output. The set $\{(1, 5), (2, 5), (3, 5)\}$ is a perfectly good function even though every output is $5$.

To test a list of ordered pairs, scan the $x$-values. If any $x$-value repeats with different $y$-values, it is not a function. In a mapping diagram — inputs in one oval, outputs in the other, arrows between — the same test reads as: every input has exactly one arrow leaving it. The diagram below is a function: input $3$ and input $4$ both land on $5$ (a shared output is fine), but no single input splits into two arrows.

Function notation: what f(x) means

Instead of writing $y = 2x + 1$, we often write $f(x) = 2x + 1$. Nothing about the rule changed — $f(x)$ is just a smarter name for $y$. The letter $f$ names the function, the $x$ in parentheses is the input, and $f(x)$ is the output the rule produces.

The payoff is that the notation carries the input with it. $f(4)$ means the output when the input is $4$: substitute $4$ everywhere you see $x$. For $f(x) = 2x + 1$, that gives $f(4) = 2(4) + 1 = 9$. One important warning: $f(x)$ is not $f$ times $x$. The parentheses here mean is a function of, not multiplication.

A table lines the inputs up with their outputs — each row is one ordered pair $(x, f(x))$. Reading the table for $f(x) = 2x + 1$: the input $3$ sits in the same row as the output $7$, which is the ordered pair $(3, 7)$.

The vertical line test

On a graph, the inputs run along the horizontal axis. So the one-output rule becomes a visual test: if any vertical line crosses the graph more than once, some input has two outputs, and the graph is not a function. If every vertical line crosses at most once, it is.

The parabola below passes the test. Slide an imaginary vertical line from left to right — at every position it touches the curve exactly once, so each input has exactly one output. A circle, by contrast, fails the test: most vertical lines through a circle hit it twice.

Worked examples

Try one yourself

Common questions