Definition:  A (Real valued) function, $f$ , (of a real variable) is a correspondence between a set of  controlling real numbers $x$ (called the domain of the function) and related controlled real number values  that are determined uniquely by the controlling numbers $x$.
Each $x$ in the domain of the function corresponds to one and only one number determined by the function, denoted:         $f (x)$.
This is sometimes denoted by writing $f: x \rightarrow  f(x)$.