Definition:  A (Real) single or multivalued function, $f$ , of a single or many real variables is a correspondence between a set of  controlling real numbers $x_1, x_2, ..., x_n$ (called the domain of the function) and related controlled real number values $y_1,y_2,...,y_k$  that are determined uniquely by the controlling numbers $x_1, x_2, ..., x_n$.
Each $n$- tuple, $(x_1, x_2, ..., x_n)$ in the domain of the function corresponds to one and only one  $k$ - tuple, $(y_1,y_2,...,y_k)$ determined by the function, denoted:         $ f (x_1, x_2, ..., x_n) = (y_1,y_2,...,y_k)$.
This is sometimes denoted by writing $f: x \rightarrow  y = f(x)$  where $x = (x_1, x_2, ..., x_n)$ and $y = (y_1,y_2,...,y_k)$.