Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other (the dependent variable, value of the function, or "output") is uniquely defined by the input.

A function associates a single output with every input element drawn from a fixed set. A function may be defined only for certain inputs, and the collection of all acceptable inputs of the function is called its domain. The set of all resulting outputs is called the range of the function. In many fields, it is also important to specify the codomain of a function, which contains the range, but need not be equal to it.

One important concept in mathematics is composition of functions: if z is a function of y and y is a function of x, then z is a function of x. This can be described informally by saying that the composite function is obtained by using the output of the first function as the input of the second one. This feature of functions distinguishes them from other mathematical constructs, such as numbers or figures.

In most mathematical fields, the terms map, mapping, and transformation are synonymous with function. However, in some contexts they may have a more specialized meaning. In particular, the term transformation usually applies to functions whose inputs and outputs are elements of the same set or more general structure. For example, we speak of linear transformations from a vector space into itself.