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 $f\colon A \to B$ . 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 $B$ .