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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the range of a function may refer to either of two closely related concepts: * The
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
of the function * The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of the function Given two
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s and , a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
between and is a (total) function (from to ) if for every in there is exactly one in such that relates to . The sets and are called
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
and codomain of , respectively. The image of is then the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of consisting of only those elements of such that there is at least one in with .


Terminology

As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article. Older books, when they use the word "range", tend to use it to mean what is now called the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
. More modern books, if they use the word "range" at all, generally use it to mean what is now called the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
. To avoid any confusion, a number of modern books don't use the word "range" at all.


Elaboration and example

Given a function :f \colon X \to Y with
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
X, the range of f, sometimes denoted \operatorname(f) or \operatorname(f), may refer to the codomain or target set Y (i.e., the set into which all of the output of f is constrained to fall), or to f(X), the image of the domain of f under f (i.e., the subset of Y consisting of all actual outputs of f). The image of a function is always a subset of the codomain of the function. As an example of the two different usages, consider the function f(x) = x^2 as it is used in
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
(that is, as a function that inputs a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
and outputs its square). In this case, its codomain is the set of real numbers \mathbb, but its image is the set of non-negative real numbers \mathbb^+, since x^2 is never negative if x is real. For this function, if we use "range" to mean ''codomain'', it refers to \mathbb; if we use "range" to mean ''image'', it refers to \mathbb^+. In many cases, the image and the codomain can coincide. For example, consider the function f(x) = 2x, which inputs a real number and outputs its double. For this function, the codomain and the image are the same (both being the set of real numbers), so the word range is unambiguous.


See also

*
Bijection, injection and surjection In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which '' arguments'' (input expressions from the domain) and '' images'' (output expressions from the codomain) are related or ''m ...
* Essential range


Notes and References


Bibliography

* * * * {{DEFAULTSORT:Range (Mathematics) Functions and mappings Basic concepts in set theory