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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.