**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 $f: x \rightarrow
f(x)$.