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A function from a set to a set associates to each element an element , in prefix notation . Often the parentheses around are omitted, for instance . Also the so-called reverse Polish notation , , is used, for instance in factorial notation .
In other words, a function from X to Y is a single-valued, total relation between X and Y, that is, for every there exists a unique element such that .
Since function concept is a special case of the concept of a relation, many notions defined for relations can be applied to functions.
The set is called the domain of , often denoted by . The target set is called the codomain of , denoted by .
A specific input in a function is called an argument and the corresponding unique in the codomain is called the function value at , or the image of under .
The concept of the image can be extended to the image of a set. If , then is the subset of the range consisting of all images of elements of and is called the image of under , denoted .
The set is called the range of . The range of is also called the image of , denoted by . In other words, .
The preimage (or inverse image) of a subset of the codomain under a function is the subset of the domain defined by
The notation is used to indicate that is a function with domain X and codomain Y. Another forms of notations are , or .
If and are two functions, then a composite function is a function defined by for all . The notation and read composed with . If function and are considered as relations then the composition of relations is again a function.
The composition of functions is always associative.
If , then the composite function , denoted , is again a function from to . Repeated composition of a function with itself is called function iteration. The following notation is used for every positive integer . Clearly, . By convention, is the identity function on the domain , defined by for each .
The identity function is neutral in the following sense: if then and .
If is a function from to then an inverse function for , denoted by , is a function from to such that if then . Because every function is a relation, the inverse relation to a function is well defined, but the inverse relation to a function may be not a function. Therefore not every function has an inverse. Moreover, inverse function is uniquely determined. If has an inverse then is called invertible.
If and are invertible functions and the inverse function exists then .
If is invertible then , , is defined by .
A restriction of a function is the result of trimming its domain to a subset . In other words, the restriction of to is the function from S to Y such that for all . If is any restriction of , we say that is an extension of .
A partial function is a partial binary relation from to that associates each element of domain with at most one (possible no) element of codomain . This means that (contrary to the definition of a function) not every element of the domain has to be associated with an element of the codomain.
A function is called (following Bourbaki)
Consider functions and , and the composition :
Let , be two functions with common domain and common codomain which is a ring with addition and multiplication . The we can define two new functions
This turns the set of all such functions into a ring. Instead of we can take any other algebraic structure to turn the set of function into an algebraic structure of the same type as .
For the history of the concept of a function visit .
Cite this web-page as:
Štefan Porubský: Function.