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The process which is symbolised by an operator is called a function.

The operand(s) of the operator can be considered to be the input(s). The output of the function is whatever the operator is defined as doing with the operand(s).

A function is in fact another name for a mapping, but while the latter term is used in the general context of set theory and abstract algebra, the term function is generally reserved for mappings between sets of numbers.

Also known as

When there is a need to distinguish between this and a partial function, a function is sometimes referred to as a total function.

When there is a need to distinguish between this and a multifunction, the term one-valued function or uniform function can be used, but this is rarely seen.

Also see

Historical Note

The term function, as used in the modern sense, was first used by Gottfried Wilhelm von Leibniz in $1694$.

The notation $\map f x$ itself appears to have originated with Leonhard Paul Euler.

He used it in two particular contexts:

particular conventional examples like trigonometric function and powers and the like
$\map y x$ for an arbitrary curve in the plane.

Up until the time of Joseph Fourier, it was accepted that a function was limited to various classes of expression: a polynomial, a finite combination of elementary functions, a power series or a trigonometric series.

Fourier made the claim that a function of arbitrary shape could be represented by a trigonometric series.

It was not until Johann Peter Gustav Lejeune Dirichlet in $1837$ that the modern definition of function was formulated:

If in any way a definite value of $y$ is determined corresponding to each value of $x$ in a given interval, then $y$ is called a function of $x$.

The concept of a mapping between arbitrary sets which are not necessarily the real or complex numbers arose in the late $19$th century.