Editing Inverse function (section)

==Generalizations==
===Partial inverses===
[[Image:Inverse square graph.svg|thumb|right|The square root of {{mvar|x}} is a partial inverse to {{math|1= ''f''(''x'') = ''x''<sup>2</sup>}}.]]
Even if a function {{mvar|f}} is not one-to-one, it may be possible to define a '''partial inverse''' of {{mvar|f}} by [[Function (mathematics)#Restrictions and extensions|restricting]] the domain. For example, the function

: <math>f(x) = x^2</math>

is not one-to-one, since {{math|1= ''x''<sup>2</sup> = (−''x'')<sup>2</sup>}}. However, the function becomes one-to-one if we restrict to the domain {{math| ''x'' ≥ 0}}, in which case

: <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.]]

The above considerations are particularly important for defining the inverses of [[trigonometric functions]]. For example, the [[sine function]] is not one-to-one, since

: <math>\sin(x + 2\pi) = \sin(x)</math>

for every real {{mvar|x}} (and more generally {{math|1= sin(''x'' + 2{{pi}}''n'') = sin(''x'')}} for every [[integer]] {{mvar|n}}). However, the sine is one-to-one on the interval
{{closed-closed|−{{sfrac|{{pi}}|2}}, {{sfrac|{{pi}}|2}}}}, and the corresponding partial inverse is called the [[arcsine]]. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −{{sfrac|{{pi}}|2}} and {{sfrac|{{pi}}|2}}. The following table describes the principal branch of each inverse trigonometric function:<ref>{{harvnb|Briggs|Cochran|2011|loc=pp. 39–42}}</ref>
{| class="wikitable" style="text-align:center"
|-
!function
!Range of usual [[principal value]]
|-
| arcsin || {{math|−{{sfrac|{{pi}}|2}} ≤ sin<sup>−1</sup>(''x'') ≤ {{sfrac|{{pi}}|2}}}}
|-
| arccos || {{math|0 ≤ cos<sup>−1</sup>(''x'') ≤ {{pi}}}}
|-
| arctan || {{math|−{{sfrac|π|2}} < tan<sup>−1</sup>(''x'') < {{sfrac|{{pi}}|2}}}}
|-
| arccot || {{math|0 < cot<sup>−1</sup>(''x'') < {{pi}}}}
|-
| arcsec || {{math|0 ≤ sec<sup>−1</sup>(''x'') ≤ {{pi}}}}
|-
| arccsc || {{math|−{{sfrac|{{pi}}|2}} ≤ csc<sup>−1</sup>(''x'') ≤ {{sfrac|{{pi}}|2}}}}
|-
|}

===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.

===Preimages===
If {{math|''f'': ''X'' → ''Y''}} is any function (not necessarily invertible), the '''preimage''' (or '''inverse image''') of an element {{math| ''y'' &isin; ''Y''}} is defined to be the set of all elements of {{mvar|X}} that map to {{mvar|y}}:

: <math>f^{-1}(y) = \left\{ x\in X : f(x) = y \right\} . </math>

The preimage of {{mvar|y}} can be thought of as the [[image (mathematics)|image]] of {{mvar|y}} under the (multivalued) full inverse of the function {{mvar|f}}.

The notion can be generalized to subsets of the range. Specifically, if {{mvar|S}} is any [[subset]] of {{mvar|Y}}, the preimage of {{mvar|S}}, denoted by <math>f^{-1}(S) </math>, is the set of all elements of {{mvar|X}} that map to {{mvar|S}}:

: <math>f^{-1}(S) = \left\{ x\in X : f(x) \in S \right\} . </math>

For example, take the function {{math|''f'': '''R''' → '''R'''; ''x'' ↦ ''x''<sup>2</sup>}}. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.

: <math>f^{-1}(\left\{1,4,9,16\right\}) = \left\{-4,-3,-2,-1,1,2,3,4\right\}</math>.

The original notion and its generalization are related by the identity <math>f^{-1}(y) = f^{-1}(\{y\}),</math> The preimage of a single element {{math| ''y'' &isin; ''Y''}} – a [[singleton set]] {{math|{''y''} }} – is sometimes called the ''[[fiber (mathematics)|fiber]]'' of {{mvar|y}}. When {{mvar|Y}} is the set of real numbers, it is common to refer to {{math|''f''<sup> −1</sup>({''y''})}} as a ''[[level set]]''.