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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, 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 the codomain or
image An SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island ...
of a function. A codomain is part of a function if is defined as a triple where is called the '' domain'' of , its ''codomain'', and its '' graph''. The set of all elements of the form , where ranges over the elements of the domain , is called the ''
image An SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island ...
'' of . The image of a function is a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of its codomain so it might not coincide with it. Namely, a function that is not
surjective In mathematics, a function (mathematics), function ''f'' from a set (mathematics), set ''X'' to a set ''Y'' is surjective (also known as onto, or a surjection), if for every Element (mathematics), element ''y'' in the codomain ''Y'' of ''f'', ther ...

has elements in its codomain for which the equation does not have a solution. A codomain is not part of a function if is defined as just a graph. For example in
set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any type of object c ...
it is desirable to permit the domain of a function to be a proper class , in which case there is formally no such thing as a triple . With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form ., p. 91 ( quote 1 [ quote 2]); , [ p. 8]; Mac Lane, in , [ p. 232]; , [ p. 91]; , [ p. 89]

# Examples

For a function :$f\colon \mathbb\rightarrow\mathbb$ defined by : $f\colon\,x\mapsto x^2,$ or equivalently $f\left(x\right)\ =\ x^2,$ the codomain of is $\textstyle \mathbb R$, but does not map to any negative number. Thus the image of is the set $\textstyle \mathbb^+_0$; i.e., the interval (mathematics), interval . An alternative function is defined thus: : $g\colon\mathbb\rightarrow\mathbb^+_0$ : $g\colon\,x\mapsto x^2.$ While and map a given to the same number, they are not, in this view, the same function because they have different codomains. A third function can be defined to demonstrate why: : $h\colon\,x\mapsto \sqrt x.$ The domain of cannot be $\textstyle \mathbb$ but can be defined to be $\textstyle \mathbb^+_0$: : $h\colon\mathbb^+_0\rightarrow\mathbb.$ The function composition, compositions are denoted : $h \circ f,$ : $h \circ g.$ On inspection, is not useful. It is true, unless defined otherwise, that the image of is not known; it is only known that it is a subset of $\textstyle \mathbb R$. For this reason, it is possible that , when composed with , might receive an argument for which no output is defined – negative numbers are not elements of the domain of , which is the square root function. Function composition therefore is a useful notion only when the ''codomain'' of the function on the right side of a composition (not its ''image'', which is a consequence of the function and could be unknown at the level of the composition) is a subset of the domain of the function on the left side. The codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. In the example, is a surjection while is not. The codomain does not affect whether a function is an injective function, injection. A second example of the difference between codomain and image is demonstrated by the linear transformations between two vector spaces – in particular, all the linear transformations from $\textstyle \mathbb^2$ to itself, which can be represented by the Matrix (mathematics), matrices with real coefficients. Each matrix represents a map with the domain $\textstyle \mathbb^2$ and codomain $\textstyle \mathbb^2$. However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with Rank (linear algebra), rank ) but many do not, instead mapping into some smaller Linear subspace, subspace (the matrices with rank or ). Take for example the matrix given by :$T = \begin 1 & 0 \\ 1 & 0 \end$ which represents a linear transformation that maps the point to . The point is not in the image of , but is still in the codomain since linear transformations from $\textstyle \mathbb^2$ to $\textstyle \mathbb^2$ are of explicit relevance. Just like all matrices, represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that does not have full rank since its image is smaller than the whole codomain.