function
A function^{} is a triplet $(f,A,B)$ where:

1.
$A$ is a set (called the domain of the function).

2.
$B$ is a set (called the codomain of the function).

3.
$f$ is a binary relation^{} between $A$ and $B$.

4.
For every $a\in A$, there exists $b\in B$ such that $(a,b)\in f$.

5.
If $a\in A$, ${b}_{1},{b}_{2}\in B$, and $(a,{b}_{1})\in f$ and $(a,{b}_{2})\in f$, then ${b}_{1}={b}_{2}$.
The triplet $(f,A,B)$ is usually written with the specialized notation $f:A\to B$. 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 $a\in A$, one denotes by $f(a)$ the unique element $b\in B$ such that $(a,b)\in f$.

•
The image of $(f,A,B)$, denoted $f(A)$, is the set
$$\{b\in B\mid f(a)=b\text{for some}a\in A\}$$ consisting of all elements of $B$ which equal $f(a)$ for some element $a\in A$. 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 $a\mapsto b$ is equivalent^{} to the statement $f(a)=b$.

•
Given two functions $f:A\to B$ and $g:B\to C$, there exists a unique function $g\circ f:A\to C$ satisfying the equation $g\circ f(a)=g(f(a))$. The function $g\circ f$ is called the composition^{} of $f$ and $g$, and a function constructed in this manner is called a composite function. Composition is associative, meaning that $h\circ (g\circ f)=(h\circ g)\circ f$ provided that either expression is defined.

•
When a function $f:A\to A$ has its domain equal to its codomain, one often writes ${f}^{n}$ for the $n$fold composition
$$\underset{n\text{times}}{\underset{\u23df}{f\circ f\circ \mathrm{\cdots}\circ f}}$$ where $n$ is any natural number^{}. Occasionally this can be confused with ordinary exponentiation (for example the function $x\mapsto (\mathrm{sin}x)(\mathrm{sin}x)$ is conventionally written as ${\mathrm{sin}}^{2}$); in such cases one usually writes ${f}^{[n]}$ to denote the $n$fold composition.
There is no universal^{} 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 theory^{}, a function is defined as a relation $f$, such that whenever $(a,b),(a,c)\in f$, 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  20130322 11:48:58 
Last modified on  20130322 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  Surjective^{} 
Related topic  Bijection 
Related topic  Relation 
Defines  domain 
Defines  codomain 
Defines  composition 
Defines  image 
Defines  range 
Defines  composite function 