Editing Implicit function (section)

==Examples==

===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}}.

===Algebraic functions===
{{main|Algebraic function}}
An '''algebraic function''' is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable {{mvar|x}} gives a solution for {{mvar|y}} of an equation

:<math>a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 \,,</math>

where the coefficients {{math|''a<sub>i</sub>''(''x'')}} are polynomial functions of {{mvar|x}}. This algebraic function can be written as the right side of the solution equation {{math|1=''y'' = ''f''(''x'')}}. Written like this,  {{mvar|f}} is a [[multi-valued function|multi-valued]] implicit function.

Algebraic functions play an important role in [[mathematical analysis]] and [[algebraic geometry]]. A simple example of an algebraic function is given by the left side of the unit circle equation:

:<math>x^2+y^2-1=0 \,. </math>

Solving for {{mvar|y}} gives an explicit solution:

:<math>y=\pm\sqrt{1-x^2} \,. </math>

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as {{math|1=''y'' = ''f''(''x'')}}, where {{mvar|f}} is the multi-valued implicit function.

While explicit solutions can be found for equations that are [[quadratic equations|quadratic]], [[cubic equation|cubic]], and [[quartic equation|quartic]] in {{mvar|y}}, the same is not in general true for [[quintic equation|quintic]] and higher degree equations, such as

:<math> y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0 \,. </math>

Nevertheless, one can still refer to the implicit solution {{math|1=''y'' = ''f''(''x'')}} involving the multi-valued implicit function {{mvar|f}}.