Editing Implicit function (section)

===Inverse functions===
A common type of implicit function is an [[inverse function]]. Not all functions have a unique inverse function. If {{mvar|g}} is a function of {{mvar|x}} that has a unique inverse, then the inverse function of {{mvar|g}}, called {{math|''g''<sup>−1</sup>}}, is the unique function giving a [[solution (mathematics)|solution]] of the equation

:<math> y=g(x) </math>

for {{mvar|x}} in terms of {{mvar|y}}. This solution can then be written as

:<math> x = g^{-1}(y) \,.</math>

Defining {{math|''g''<sup>−1</sup>}} as the inverse of {{mvar|g}} is an implicit definition. For some functions {{mvar|g}}, {{math|''g''<sup>−1</sup>(''y'')}} can be written out explicitly as a [[closed-form expression]] — for instance, if {{math|1=''g''(''x'') = 2''x'' − 1}}, then {{math|1=''g''<sup>−1</sup>(''y'') = {{sfrac|1|2}}(''y'' + 1)}}. However, this is often not possible, or only by introducing a new notation (as in the [[product log]] example below).

Intuitively, an inverse function is obtained from {{mvar|g}} by interchanging the roles of the dependent and independent variables.

'''Example:''' The [[product log]] is an implicit function giving the solution for {{mvar|x}} of the equation {{math|1=''y'' − ''xe''<sup>''x''</sup> = 0}}.