Inverse function: Difference between revisions

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====Left inverses====

If {{math|''f'': ''X'' → ''Y''}}, a '''left inverse''' for {{mvar|f}} (or ''[[retract (category theory)|retraction]]'' of {{mvar|f}} ) is a function {{math| ''g'': ''Y'' → ''X''}} such that composing {{mvar|f}} with {{mvar|g}} from the left gives the identity function<ref>{{Citationcite neededbook|datelast1=FebruaryDummit|last2=Foote|title=Abstract 2021Algebra}}:</ref> <math display="block">g \circ f = \operatorname{id}_X\text{.}</math> That is, the function {{mvar|g}} satisfies the rule

: If {{math|''f''(''x''){{=}}''y''}}, then {{math|''g''(''y''){{=}}''x''}}.

<math display="block">g \circ f = \operatorname{id}_X . </math>

Thus,The function {{mvar|g}} must equal the inverse of {{mvar|f}} on the image of {{mvar|f}}, but may take any values for elements of {{mvar|Y}} not in the image.

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

A function {{mvar|f}} with nonempty ___domain is injective if and only if it has a left inverse.<ref>{{cite book|last=Mac&nbsp;Lane|first=Saunders|title=Categories for the Working Mathematician}}</ref> An elementary proof runs as follows:

: If <math>f(x) = y</math>, then <math>g(y) = x .</math>

:* If {{mvar|g}} is the left inverse of {{mvar|f}}, then {{mvar|f}} is injective. Ifand {{math|1=''f''(''x'') = ''f''(''y'')}}, then <{{math>|1=''g''(''f''(''x'')) = ''g''(''f''(''y'')) = ''x'' = ''y</math>''}}.

:* <p>If nonempty {{math|''f'': ''X'' → ''Y''}} is injective, {{mvar|f}} either is the empty function ({{math|1=''X'' = ∅}}) or hasconstruct a left inverse {{math|''g'': ''Y'' → ''X''}} ({{math|''X'' ≠ ∅)}}, which can be constructed as follows: for all {{math|''y'' ∈ ''Y''}}, if {{mvar|y}} is in the image of {{mvar|f}}, then (there exists {{math|''x'' ∈ ''X''}} such that {{math|1=''f''(''x'') = ''y''}}),. let Let {{math|1=''g''(''y'') = ''x''}}; ({{mvar|x}}this definition is unique because {{mvar|f}} is injective);. otherwiseOtherwise, let {{math|''g''(''y'')}} be an arbitrary element of {{mvar|X}}. </p><p>For all {{math|''x'' ∈ ''X''}}, {{math|''f''(''x'')}} is in the image of {{mvar|f}}. By construction, so {{math|1=''g''(''f''(''x'')) = ''x''}} by above, sothe {{mvar|g}}condition isfor a left inverse of {{mvar|f}}.</p>

In classical mathematics, every injective function {{mvar|f}} with a nonempty ___domain necessarily has a left inverse; however, this may fail in [[constructive mathematics]]. For instance, a left inverse of the inclusion {{math|{0,1} → '''R'''}} of the two-element set in the reals violates [[indecomposability (constructive mathematics)|indecomposability]] by giving a [[Retract (category theory)|retraction]] of the real line to the set {{math|{0,1{{)}}}}.<ref>{{Citationcite neededbook|datelast=FebruaryFraenkel|title=Abstract Set 2021Theory}}</ref>

Thus, {{mvar|g}} must equal the inverse of {{mvar|f}} on the image of {{mvar|f}}, but may take any values for elements of {{mvar|Y}} not in the image.

A function {{mvar|f}} is injective if and only if it has a left inverse or is the empty function.{{Citation needed|date=February 2021}}

: If {{mvar|g}} is the left inverse of {{mvar|f}}, then {{mvar|f}} is injective. If {{math|1=''f''(''x'') = ''f''(''y'')}}, then <math>g(f(x)) = g(f(y)) = x = y</math>.

: If {{math|''f'': ''X'' → ''Y''}} is injective, {{mvar|f}} either is the empty function ({{math|1=''X'' = ∅}}) or has a left inverse {{math|''g'': ''Y'' → ''X''}} ({{math|''X'' ≠ ∅)}}, which can be constructed as follows: for all {{math|''y'' ∈ ''Y''}}, if {{mvar|y}} is in the image of {{mvar|f}} (there exists {{math|''x'' ∈ ''X''}} such that {{math|1=''f''(''x'') = ''y''}}), let {{math|1=''g''(''y'') = ''x''}} ({{mvar|x}} is unique because {{mvar|f}} is injective); otherwise, let {{math|''g''(''y'')}} be an arbitrary element of {{mvar|X}}. For all {{math|''x'' ∈ ''X''}}, {{math|''f''(''x'')}} is in the image of {{mvar|f}}, so {{math|1=''g''(''f''(''x'')) = ''x''}} by above, so {{mvar|g}} is a left inverse of {{mvar|f}}.

In classical mathematics, every injective function {{mvar|f}} with a nonempty ___domain necessarily has a left inverse; however, this may fail in [[constructive mathematics]]. For instance, a left inverse of the inclusion {{math|{0,1} → '''R'''}} of the two-element set in the reals violates [[indecomposability (constructive mathematics)|indecomposability]] by giving a [[Retract (category theory)|retraction]] of the real line to the set {{math|{0,1{{)}}}}.{{Citation needed|date=February 2021}}

====Right inverses====