Implicit function

In mathematics, an implicit equation is a relation of the form R ( x 1 , … , x n ) = 0 {displaystyle R(x_{1},ldots ,x_{n})=0} , where R {displaystyle R} is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x 2 + y 2 − 1 = 0 {displaystyle x^{2}+y^{2}-1=0} . In mathematics, an implicit equation is a relation of the form R ( x 1 , … , x n ) = 0 {displaystyle R(x_{1},ldots ,x_{n})=0} , where R {displaystyle R} is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x 2 + y 2 − 1 = 0 {displaystyle x^{2}+y^{2}-1=0} . An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments).:204–206 Thus, an implicit function for y {displaystyle y} in the context of the unit circle is defined implicitly by x 2 + f ( x ) 2 − 1 = 0 {displaystyle x^{2}+f(x)^{2}-1=0} . This implicit equation defines f {displaystyle f} as a function of x {displaystyle x} only if − 1 ≤ x ≤ 1 {displaystyle -1leq xleq 1} and one considers only non-negative (or non-positive) values for the values of the function. The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function. A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If g is a function of x that has a unique inverse, then the inverse function of g, called g−1, is the unique function giving a solution of the equation for x in terms of y. This solution can then be written as Defining g − 1 {displaystyle g^{-1}} as the inverse of g {displaystyle g} is an implicit definition. For some functions g, g − 1 ( y ) {displaystyle g^{-1}(y)} can be written out explicitly as a closed-form expression — for instance, if g ( x ) = 2 x − 1 , {displaystyle g(x)=2x-1,} then g − 1 ( y ) = 1 2 ( y + 1 ) {displaystyle g^{-1}(y)={ frac {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 g by interchanging the roles of the dependent and independent variables. Example The product log is an implicit function giving the solution for x of the equation y − xex = 0.