Function (mathematics)

The mathematical concept of a function (also called a mapping or map) 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 or image 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 function composition: 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 operator, operation, and transformation are synonymous with function. However, in some contexts they may have a more specialized meaning. In particular, they often apply to functions whose inputs and outputs are elements of the same set. For example, we speak of linear operators on a vector space, which are linear transformations from the vector space into itself.

Special classes of function

• An injective function f has the property that if $x_{1}\neq x_{2}$ then $f(x_{1})\neq f(x_{2})$;
• A surjective function f has the property that for every y in the codomain there exists an x in the domain such that $f(x)=y$;
• A bijective function is one which is both surjective and injective.

Functions in set theory

In set theory, functions are regarded as a special class of relation. A relation between sets X and Y is a subset of the Cartesian product, $R\subseteq X\times Y$. We say that a relation R is functional if it satisfies the condition that every $x\in X$ occurs in exactly one pair $(x,y)\in R$. In this case R defines a function with domain X and codomain Y. We then define the value of the function at x to be that unique y. We thus identify a function with its graph.

Associated sets

Let f:X → Y be a function with domain X and codomain Y. The image of a subset A of X is $f[A]=\{f(x):x\in A\}$; the image of f is the image of X under f. The pre-image of a subset B of Y is $f^{-1}[B]=\{x\in X:f(x)\in B\}$. The fibre of f over a point y in Y is the preimage of the singleton {y}. The kernel of f is the equivalence relation on X for which the equivalence classes are the fibres of f.

Associated functions

If f is a function from a set X to a set Y, there are several functions associated with f.

If S is a subset of X, the restriction of f to S is the function from S to Y which is given by applying f only to elements of S. The restriction may have different properties to the original. Consider the function $f:x\mapsto x^{2}$ from the real numbers R to R. The restriction of f to the positive real numbers is injective, whereas f is not.

The push-forward of f is the function $f_{\vdash }$ from the power set of X to that of Y which maps a subset A of X to its image in Y:

$f_{\vdash }(A)=\{f(x):x\in A\}.\,$

An alternative notation for $f_{\vdash }(A)$ is $f[A]$ (note the square brackets).

The pull-back of f is the function $f^{\dashv }$ from the power set of Y to the power set of X which maps a subset B of Y to its pre-image in X:

$f^{\dashv }(B)=\{x\in X:f(x)\in B\}.\,$

An alternative notation for $f^{\dashv }(B)$ is $f^{-1}[B]$ (note the square brackets). Pull-back is a generalised form of inverse, and makes sense whether or not f is an invertible function.