# Implicit function

In mathematics, an implicit equation is a relation of the form ${\displaystyle R(x_{1},\dots ,x_{n})=0,}$ where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is ${\displaystyle x^{2}+y^{2}-1=0.}$

An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments.[1]:204–206 For example, the equation ${\displaystyle x^{2}+y^{2}-1=0}$ of the unit circle defines y as an implicit function of x if −1 ≤ x ≤ 1, and one restricts y to nonnegative values.

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.