Inverse function: Difference between revisions

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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 [[function (mathematics)|function]] that undoes the operation of {{Mvar|f}}. The inverse of {{Mvar|f}} exists [[if and only if]] {{Mvar|f}} is [[Bijection|bijective]], and if it exists, is denoted by <math>f^{-1} .</math>
 
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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: 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>{{Cite book |last=Loehr |first=Nicholas A. |url=https://wwwbooks.google.co.ilcom/books/edition/An_Introduction_to_Mathematical_Proofs/mGUIEQAAQBAJ?hlid=iw&gbpv=1mGUIEQAAQBAJ&pg=PA272&printsec=frontcover |title=An Introduction to Mathematical Proofs |date=2019-11-20 |publisher=CRC Press |isbn=978-1-000-70962-9 |language=en}}</ref>
 
====Two-sided inverses====