Editing Inverse function (section)

====Right inverses====
[[File:Right inverse with surjective function.svg|thumb|Example of '''right inverse''' with non-injective, surjective function]]
A '''right inverse''' for {{mvar|f}} (or ''[[section (category theory)|section]]'' of {{mvar|f}} ) is a function {{math| ''h'': ''Y'' → ''X''}} such that

: <math>f \circ h = \operatorname{id}_Y . </math>

That is, the function {{mvar|h}} satisfies the rule

: If <math>\displaystyle h(y) = x</math>, then <math>\displaystyle f(x) = y .</math>

Thus, {{math|''h''(''y'')}} may be any of the elements of {{mvar|X}} that map to {{mvar|y}} under {{mvar|f}}.

A function {{mvar|f}} has a right inverse if and only if it is [[surjective function|surjective]] (though constructing such an inverse in general requires the [[axiom of choice]]).

: 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://books.google.com/books?id=mGUIEQAAQBAJ&pg=PA272 |title=An Introduction to Mathematical Proofs |date=2019-11-20 |publisher=CRC Press |isbn=978-1-000-70962-9 |language=en}}</ref>