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.

Also see

• Definition:Mapping

• Definition:Real Function
• Definition:Real-Valued Function

• Definition:Complex Function
• Definition:Complex-Valued Function

• Definition:Truth Function
• Definition:Propositional Function

• Definition:Partial Function
• Definition:Multifunction

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.

Sources

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
• 1937: Richard Courant: Differential and Integral Calculus: Volume $\text { I }$ (2nd ed.) ... (next): Chapter $\text I$: Introduction
• 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (next): Chapter $\text I$: Differentiation: Functions
• 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (next): Chapter $\text I$: Functions and Limits: $\S 2$: Functions
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.1$. Mappings: Example $45$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.1$: Motivation
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.1$
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $6$: Curves and Coordinates: Functions