Editing Inverse function (section)

==Properties==
Since a function is a special type of [[binary relation]], many of the properties of an inverse function correspond to properties of [[converse relation]]s.

===Uniqueness===
If an inverse function exists for a given function {{mvar|f}}, then it is unique.<ref name="Wolf72">{{harvnb|Wolf|1998|loc=p. 208, Theorem 7.2}}</ref> This follows since the inverse function must be the converse relation, which is completely determined by {{mvar|f}}.

===Symmetry===
There is a symmetry between a function and its inverse. Specifically, if {{mvar|f}} is an invertible function with domain {{mvar|X}} and codomain {{mvar|Y}}, then its inverse {{math|''f''<sup> −1</sup>}} has domain {{mvar|Y}} and image {{mvar|X}}, and the inverse of {{math|''f''<sup> −1</sup>}} is the original function {{mvar|f}}. In symbols, for functions {{math|''f'':''X'' → ''Y''}}  and {{math|''f''<sup>−1</sup>:''Y'' → ''X''}},<ref name=Wolf72 />

:<math>f^{-1}\circ f = \operatorname{id}_X </math> and <math> f \circ f^{-1} = \operatorname{id}_Y.</math>

This statement is a consequence of the implication that for {{mvar|f}} to be invertible it must be bijective. The [[involution (mathematics)|involutory]] nature of the inverse can be concisely expressed by<ref>{{harvnb|Smith|Eggen|St. Andre|2006|loc=pg. 141 Theorem 3.3(a)}}</ref> 
:<math>\left(f^{-1}\right)^{-1} = f.</math>

[[Image:Composition of Inverses.png|thumb|right|240px|The inverse of {{math| ''g'' ∘ ''f'' }} is {{math| ''f''<sup> −1</sup> ∘ ''g''<sup> −1</sup>}}.]]
The inverse of a composition of functions is given by<ref>{{harvnb|Lay|2006|loc=p. 71, Theorem 7.26}}</ref> 
:<math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}.</math>
Notice that the order of {{mvar|g}} and {{mvar|f}} have been reversed; to undo {{mvar|f}} followed by {{mvar|g}}, we must first undo {{mvar|g}}, and then undo {{mvar|f}}.

For example, let {{math|1= ''f''(''x'') = 3''x''}} and let {{math|1= ''g''(''x'') = ''x'' + 5}}. Then the composition {{math| ''g'' ∘ ''f''}} is the function that first multiplies by three and then adds five,

: <math>(g \circ f)(x) = 3x + 5.</math>

To reverse this process, we must first subtract five, and then divide by three,

: <math>(g \circ f)^{-1}(x) = \tfrac13(x - 5).</math>

This is the composition
{{math| (''f''<sup> −1</sup> ∘ ''g''<sup> −1</sup>)(''x'')}}.

===Self-inverses===
If {{mvar|X}} is a set, then the [[identity function]] on {{mvar|X}} is its own inverse:

: <math>{\operatorname{id}_X}^{-1} = \operatorname{id}_X.</math>

More generally, a function {{math| ''f'' : ''X'' → ''X''}} is equal to its own inverse, if and only if the composition {{math| ''f'' ∘ ''f''}} is equal to {{math|id<sub>''X''</sub>}}. Such a function is called an [[Involution (mathematics)|involution]].

===Graph of the inverse===
[[Image:Inverse Function Graph.png|thumb|right|The graphs of {{math|1= ''y'' = ''f''(''x'') }} and {{math|1= ''y'' = ''f''<sup> −1</sup>(''x'')}}. The dotted line is {{math|1= ''y'' = ''x''}}.]]
If  {{mvar|f}} is invertible, then the graph of the function

: <math>y = f^{-1}(x)</math>

is the same as the graph of the equation

: <math>x = f(y) .</math>

This is identical to the equation {{math|1= ''y'' = ''f''(''x'')}} that defines the graph of {{mvar|f}}, except that the roles of {{mvar|x}} and {{mvar|y}} have been reversed. Thus the graph of {{math|''f''<sup> −1</sup>}} can be obtained from the graph of {{mvar|f}} by switching the positions of the {{mvar|x}} and {{mvar|y}} axes. This is equivalent to [[Reflection (mathematics)|reflecting]] the graph across the line
{{math|1= ''y'' = ''x''}}.<ref>{{harvnb|Briggs|Cochran|2011|loc=pp. 28–29}}</ref><ref name=":2" />

===Inverses and derivatives===
By the [[inverse function theorem]], a [[continuous function]] of a single variable <math>f\colon A\to\mathbb{R}</math> (where <math>A\subseteq\mathbb{R}</math>) is invertible on its range (image) if and only if it is either strictly [[monotonic function|increasing or decreasing]] (with no local [[maxima and minima|maxima or minima]]). For example, the function

: <math>f(x) = x^3 + x</math>

is invertible, since the [[derivative]]
{{math|1= ''f&prime;''(''x'') = 3''x''<sup>2</sup> + 1 }} is always positive.

If the function {{mvar|f}} is [[Differentiable function|differentiable]] on an interval {{mvar|I}} and {{math| ''f&prime;''(''x'') ≠ 0}} for each {{math|''x'' ∈ ''I''}}, then the inverse {{math|''f''<sup> −1</sup>}} is differentiable on {{math|''f''(''I'')}}.<ref>{{harvnb|Lay|2006|loc=p. 246, Theorem 26.10}}</ref>  If {{math|1= ''y'' = ''f''(''x'')}}, the derivative of the inverse is given by the inverse function theorem,

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

Using [[Leibniz's notation]] the formula above can be written as

: <math>\frac{dx}{dy} = \frac{1}{dy / dx}. </math>

This result follows from the [[chain rule]] (see the article on [[inverse functions and differentiation]]).

The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable [[real multivariable function|multivariable function]] {{math| ''f '': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} is invertible in a neighborhood of a point {{mvar|p}} as long as the [[Jacobian matrix and determinant|Jacobian matrix]] of {{mvar|f}} at {{mvar|p}} is [[invertible matrix|invertible]]. In this case, the Jacobian of {{math|''f''<sup> −1</sup>}} at {{math|''f''(''p'')}} is the [[matrix inverse]] of the Jacobian of {{mvar|f}} at {{mvar|p}}.