# Definition:Function

## Definition

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.

## Historical Note

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

The notation $f \left({x}\right)$ itself appears to have originated with Leonhard Paul Euler.

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.