Implicit function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an implicit equation is a <a href="/facts/Relation_(mathematics)/HjblDaMy">relation</a> of the form 
 
 
 
 R
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 =
 0
 ,
 
 
 {\displaystyle R(x_{1},\dots ,x_{n})=0,}
 
 where R is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of several variables (often a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a>). For example, the implicit equation of the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> is 
 
 
 
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 −
 1
 =
 0.
 
 
 {\displaystyle x^{2}+y^{2}-1=0.}

An implicit function is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that is defined by an implicit equation, that relates one of the variables, considered as the <a href="/facts/Value_(mathematics)/mGYQu2xt">value</a> of the function, with the others considered as the <a href="/facts/Argument_of_a_function/QkuPc28I">arguments</a>.: 204–206  For example, the equation 
 
 
 
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 −
 1
 =
 0
 
 
 {\displaystyle x^{2}+y^{2}-1=0}
 
 of the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> defines y as an implicit function of x if −1 ≤ x ≤ 1, and y is restricted to nonnegative values.
The <a href="/facts/Implicit_function_theorem/r4h0RpWw">implicit function theorem</a> provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero <a href="/facts/Multivariable_function/cyujR2q2">multivariable functions</a> that are <a href="/facts/Continuously_differentiable/jQqTgLk1">continuously differentiable</a>.

Implicit function open-in-new

Implicit function