Editing Inverse function (section)

===Left and right inverses===
[[Function composition]] on the left and on the right need not coincide.  In general, the conditions 
# "There exists {{mvar|g}} such that {{math|''g''(''f''(''x'')){{=}}''x''}}" and 
# "There exists {{mvar|g}} such that {{math|''f''(''g''(''x'')){{=}}''x''}}"
imply different properties of {{mvar|f}}.  For example, let {{math|''f'': '''R''' → {{closed-open|0, ∞}}}} denote the squaring map, such that {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} for all {{mvar|x}} in {{math|'''R'''}}, and let  {{math|{{mvar|g}}: {{closed-open|0, ∞}} → '''R'''}} denote the square root map, such that {{math|''g''(''x'') {{=}} }}{{radic|{{mvar|x}}}} for all {{math|''x'' ≥ 0}}. Then {{math|1=''f''(''g''(''x'')) = ''x''}} for all {{mvar|x}} in {{closed-open|0, ∞}}; that is, {{mvar|g}} is a right inverse to {{mvar|f}}. However, {{mvar|g}} is not a left inverse to {{mvar|f}}, since, e.g., {{math|1=''g''(''f''(−1)) = 1 ≠ −1}}.

====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>{{cite book|last1=Dummit|last2=Foote|title=Abstract Algebra}}</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''}}.

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.

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 {{mvar|g}} is the left inverse of {{mvar|f}}, and {{math|1=''f''(''x'') = ''f''(''y'')}}, then {{math|1=''g''(''f''(''x'')) = ''g''(''f''(''y'')) = ''x'' = ''y''}}.
* <p>If nonempty {{math|''f'': ''X'' → ''Y''}} is injective, construct a left inverse {{math|''g'': ''Y'' → ''X''}} 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 {{math|1=''g''(''y'') = ''x''}}; this definition is unique because {{mvar|f}} is injective.  Otherwise, 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, {{math|1=''g''(''f''(''x'')) = ''x''}}, the condition for a left inverse.</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 map|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>{{cite journal|last=Fraenkel|title=Abstract Set Theory|journal=Nature |year=1954 |volume=173 |issue=4412 |page=967 |doi=10.1038/173967a0 |bibcode=1954Natur.173..967C |s2cid=7735523 |doi-access=free }}</ref>

====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>

====Two-sided inverses====
An inverse that is both a left and right inverse (a '''two-sided inverse'''), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called '''the inverse'''.

: If <math>g</math> is a left inverse and <math>h</math> a right inverse of <math>f</math>, for all <math>y \in Y</math>, <math>g(y) = g(f(h(y)) = h(y)</math>.

A function has a two-sided inverse if and only if it is bijective.
: A bijective function {{mvar|f}} is injective, so it has a left inverse (if {{mvar|f}} is the empty function, <math>f \colon \varnothing \to \varnothing</math> is its own left inverse). {{mvar|f}} is surjective, so it has a right inverse. By the above, the left and right inverse are the same.
: If {{mvar|f}} has a two-sided inverse {{mvar|g}}, then {{mvar|g}} is a left inverse and right inverse of {{mvar|f}}, so {{mvar|f}} is injective and surjective.