A function 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 relation between $A$ and $B$.

4. 4.

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

5. 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 $f\colon A\to B$. 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 $f\colon A\to B$ and $g\colon B\to C$, there exists a unique function $g\circ f\colon A\to C$ 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 $f\colon A\to A$ has its domain equal to its codomain, one often writes $f^{n}$ for the $n$-fold composition

  $$\underbrace{f\circ f\circ\cdots\circ f}_{{n\text{ times}}}$$

  where $n$ is any natural number. Occasionally this can be confused with ordinary exponentiation (for example the function $x\mapsto(\sin x)(\sin x)$ is conventionally written as $\sin^{2}$); 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 $B$.