We already met the idea of a function — a rule that turns each input into exactly one output. Now we need a tidy way to write that rule down. That shorthand is called function notation.

Instead of saying "the rule that doubles a number and adds one", we give the rule a name, usually f, and write:

Read this out loud as "f of x equals two x plus one". The x in the brackets is the input, and the whole expression f(x) stands for the output the rule produces.

The brackets are not multiplication

This is the one trap to watch. f(x) does not mean f times x. The brackets are a kind of slot: whatever you put inside them is fed into the rule.

So f(3) means "run the rule with x = 3". We take the formula, replace every x with 3, and simplify:

Use the machine below: pick an input on the slider and watch the rule f(x) = 2x + 1 substitute it in and report the output f(x).

Inputs and outputs on a graph

Each evaluation f(a) = b is also a point on the graph of the function: the input a runs along the bottom, and the output b is the height of the curve there. Drag the input below and watch the point (x,\,f(x)) ride along the line.

The name is just a label

The letter f is only a label — we can use any name we like, and often the name hints at what the function describes. A height function might be h(t), a cost function C(n). They all work the same way: feed the input inside the brackets, run the rule, read the output.

Khan Academy works through evaluating with function notation here: