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[[Image:Inverse Function.png|thumb|right|A function {{mvar|f}} and its inverse {{math|''f''<sup> −1</sup>}}. Because {{mvar|f}} maps {{mvar|a}} to 3, the inverse {{math|''f''<sup> −1</sup>}} maps 3 back to {{mvar|a}}.]]
{{Functions}}
In [[mathematics]], the '''inverse function''' of a [[Function (mathematics)|function]] {{Mvar|f}} (also called the '''inverse''' of {{Mvar|f}}) is a
For a function <math>f\colon X\to Y</math>, its inverse <math>f^{-1}\colon Y\to X</math> admits an explicit description: it sends each element <math>y\in Y</math> to the unique element <math>x\in X</math> such that {{Math|1=''f''(''x'') = ''y''}}.
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: <math>f^{-1}(y) = \sqrt{y} . </math>
(If we instead restrict to the ___domain {{math| ''x'' ≤ 0}}, then the inverse is the negative of the square root of {{mvar|y}}.)
===Full inverses=== [[File:Inversa d'una cúbica gràfica.png|right|thumb|The inverse of this [[cubic function]] has three branches.]] Alternatively, there is no need to restrict the ___domain if we are content with the inverse being a [[multivalued function]]: : <math>f^{-1}(y) = \pm\sqrt{y} . </math>
Sometimes, this multivalued inverse is called the '''full inverse''' of {{mvar|f}}, and the portions (such as {{sqrt|{{mvar|x}}}} and −{{sqrt|{{mvar|x}}}}) are called ''branches''. The most important branch of a multivalued function (e.g. the positive square root) is called the ''[[principal branch]]'', and its value at {{mvar|y}} is called the ''principal value'' of {{math|''f''<sup> −1</sup>(''y'')}}.
For a continuous function on the real line, one branch is required between each pair of [[minima and maxima|local extrema]]. For example, the inverse of a [[cubic function]] with a local maximum and a local minimum has three branches (see the adjacent picture).
===Trigonometric inverses===
[[Image:Gràfica del arcsinus.png|right|thumb|The [[arcsine]] is a partial inverse of the [[sine]] function.]]
: <math>\sin(x + 2\pi) = \sin(x)</math>
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: If {{mvar|h}} is the right inverse of {{mvar|f}}, then {{mvar|f}} is surjective. For all <math>y \in Y</math>, there is <math>x = h(y)</math> such that <math>f(x) = f(h(y)) = y</math>.
: If {{mvar|f}} is surjective, {{mvar|f}} has a right inverse {{mvar|h}}, which can be constructed as follows: for all <math>y \in Y</math>, there is at least one <math>x \in X</math> such that <math>f(x) = y</math> (because {{mvar|f}} is surjective), so we choose one to be the value of {{math|''h''(''y'')}}.<ref>{{
====Two-sided inverses====
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