Input, Process, Output

A function takes an input, applies a rule, and produces exactly one output. Feed 3 into f(x) = 2x + 1 and out comes 7. Feed in -4, out comes -7. A function is a machine: the same input always produces the same output, guaranteed.

The notation f(x) does not mean f times x. It means "the function f, evaluated at x." The parentheses signal an input, not multiplication. Leonhard Euler introduced this f(x) notation in 1734, and it is still the standard across all of mathematics.