Editing Surjective function (section)

==Examples==
[[File:Codomain2.SVG|right|thumb|250px|'''A non-surjective function''' from [[domain of a function|domain]] ''X'' to [[codomain]] ''Y''. The smaller yellow oval inside ''Y'' is the [[Image (mathematics)|image]] (also called [[range of a function|range]]) of ''f''. This function is '''not''' surjective, because the image does not fill the whole codomain. In other words, ''Y'' is colored in a two-step process: First, for every ''x'' in ''X'', the point ''f''(''x'') is colored yellow; Second, all the rest of the points in ''Y'', that are not yellow, are colored blue. The function ''f'' would be surjective only if there were no blue points.]]
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* For any set ''X'', the [[identity function]] id<sub>''X''</sub> on ''X'' is surjective.
* The function {{math|''f'' : '''Z''' → {0, 1}<nowiki/>}} defined by ''f''(''n'') = ''n'' '''[[Modular arithmetic|mod]]''' 2 (that is, [[even number|even]] [[integer]]s are mapped to 0 and [[odd number|odd]] integers to 1) is surjective.
* The function {{math|''f'' : '''R''' → '''R'''}} defined by ''f''(''x'') = 2''x'' + 1 is surjective (and even [[bijective function|bijective]]), because for every [[real number]] ''y'', we have an ''x'' such that ''f''(''x'') = ''y'': such an appropriate ''x'' is (''y'' − 1)/2.
* The function {{math|''f'' : '''R''' → '''R'''}} defined by ''f''(''x'') = ''x''<sup>3</sup> − 3''x'' is surjective, because the pre-image of any [[real number]] ''y'' is the solution set of the cubic polynomial equation ''x''<sup>3</sup> − 3''x'' − ''y'' = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not [[injective function|injective]] (and hence not [[bijective function|bijective]]), since, for example, the pre-image of ''y'' = 2 is {''x'' = −1, ''x'' = 2}. (In fact, the pre-image of this function for every ''y'',  −2 ≤ ''y'' ≤ 2 has more than one element.)
* The function {{math|''g'' : '''R''' → '''R'''}} defined by {{Nowrap begin}}''g''(''x'') = ''x''<sup>2</sup>{{Nowrap end}} is ''not'' surjective, since there is no real number ''x'' such that {{Nowrap begin}}''x''<sup>2</sup> = −1{{Nowrap end}}. However, the function {{math|''g'' : '''R''' → '''R'''{{sub|≥0}}}} defined by {{math|1=''g''(''x'') = ''x''<sup>2</sup>}} (with the restricted codomain) ''is'' surjective, since for every ''y'' in the nonnegative real codomain ''Y'', there is at least one ''x'' in the real domain ''X'' such that {{Nowrap begin}}''x''<sup>2</sup> = ''y''{{Nowrap end}}.
* The [[natural logarithm]] function {{math|ln : <nowiki>(0, +∞)</nowiki> → '''R'''}} is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the [[exponential function]], if defined with the set of real numbers as the domain and the codomain, is not surjective (as its range is the set of positive real numbers). 
* The [[matrix exponential]] is not surjective when seen as a map from the space of all ''n''×''n'' [[matrix (mathematics)|matrices]] to itself. It is, however, usually defined as a map from the space of all ''n''×''n'' matrices to the [[general linear group]] of degree ''n'' (that is, the [[group (mathematics)|group]] of all ''n''×''n'' [[invertible matrix|invertible matrices]]). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.
* The [[projection (set theory)|projection]] from a [[cartesian product]] {{math|''A'' × ''B''}} to one of its factors is surjective, unless the other factor is empty.
* In a 3D video game, vectors are projected onto a 2D flat screen by means of a surjective function.

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