A functionMathworldPlanetmath is a triplet (f,A,B) where:

  1. 1.

    A is a set (called the domain of the function).

  2. 2.

    B is a set (called the codomain of the function).

  3. 3.

    f is a binary relationMathworldPlanetmath between A and B.

  4. 4.

    For every aA, there exists bB such that (a,b)f.

  5. 5.

    If aA, b1,b2B, and (a,b1)f and (a,b2)f, then b1=b2.

The triplet (f,A,B) is usually written with the specialized notation f:AB. This notation visually conveys the fact that f maps elements of A into elements of B.

Other standard notations for functions are as follows:

  • For aA, one denotes by f(a) the unique element bB such that (a,b)f.

  • The image of (f,A,B), denoted f(A), is the set

    {bBf(a)=b for some aA}

    consisting of all elements of B which equal f(a) for some element aA. Note that, by abuse of notation, the set f(A) is almost always called the image of f, rather than the image of (f,A,B).

  • In cases where the function f is clear from context, the notation ab is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the statement f(a)=b.

  • Given two functions f:AB and g:BC, there exists a unique function gf:AC satisfying the equation gf(a)=g(f(a)). The function gf is called the compositionMathworldPlanetmath of f and g, and a function constructed in this manner is called a composite function. Composition is associative, meaning that h(gf)=(hg)f provided that either expression is defined.

  • When a function f:AA has its domain equal to its codomain, one often writes fn for the n-fold composition

    fffn times

    where n is any natural numberMathworldPlanetmath. Occasionally this can be confused with ordinary exponentiation (for example the function x(sinx)(sinx) is conventionally written as sin2); in such cases one usually writes f[n] to denote the n-fold composition.

There is no universalPlanetmathPlanetmath agreement as to the definition of the range of a function. Some authors define the range of a function to be equal to the codomain, and others define the range of a function to be equal to the image.

Remark. In set theoryMathworldPlanetmath, a function is defined as a relation f, such that whenever (a,b),(a,c)f, then b=c. Notice that the sets A,B are not specified in advance, unlike the defintion given in the beginning of the article. The domain and range of the function f is the domain and range of f as a relation. Using this definition of a function, we may recapture the defintion at the top of the entry by saying that a function f maps from a set A into a set B, if the domain of f is A, and the range of f is a subset of B.

Title function
Canonical name Function
Date of creation 2013-03-22 11:48:58
Last modified on 2013-03-22 11:48:58
Owner djao (24)
Last modified by djao (24)
Numerical id 23
Author djao (24)
Entry type Definition
Classification msc 03E20
Classification msc 44A20
Classification msc 33E20
Classification msc 30D15
Synonym map
Related topic Mapping
Related topic InjectiveFunction
Related topic SurjectivePlanetmathPlanetmath
Related topic Bijection
Related topic Relation
Defines domain
Defines codomain
Defines composition
Defines image
Defines range
Defines composite function