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. 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 y is restricted to nonnegative values.

The implicit function theorem provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero multivariable functions that are continuously differentiable.

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 as the inverse of g is an implicit definition. For some functions g, g⁻¹(y) can be written out explicitly as a closed-form expression — for instance, if g(x) = 2x − 1, then g⁻¹(y) = ⁠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 − xe^x = 0.