function notation

Function notation is the compact way we write “the output of f at x” as f(x), read aloud as “f of x.” The name of the machine is f, the thing in the parentheses is the input, and the whole symbol f(x) names the output. It is shorthand, not multiplication: f(x) does not mean f times x.

Its great convenience is that you can name the input directly. Writing f(3) means “run the machine on 3,” and to evaluate it you substitute 3 wherever x appears in the rule. You can also use any letter for the machine — g(x), h(t), P(n) — which lets you juggle several functions in one problem without confusion.

Notation like this also makes transformations crisp to state. f(x) + 2 says “take the output and add 2,” while f(x + 2) says “shift the input by 2 first.” These look similar but mean different things, and the parentheses tell you exactly which.