# Definition:Function

## Definition

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

The operand(s) of the operation can be considered to be the input(s).

The output of the function is whatever the operation 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.

## Examples

### Velocity of Falling Body

The velocity of a body in free fall towards Earth from a given point is a function of time.

### Pressure of Gas

The pressure of a gas at a constant temperature is a function of its volume.

### Time Period of Pendulum

The time period of a pendulum is a function of its length.

## 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.