In mathematics, an implicit equation is a relation of the form where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is 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 of the unit circle defines as an implicit function of if , and 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. Examples Inverse functions A common type of implicit function is an inverse function. Not all functions have a unique inverse function. If is a function of that has a unique inverse, then the inverse function of , called , is the unique function giving a solution of the equation for in terms of . This solution can then be written as Defining as the inverse of is an implicit definition. For some functions , can be written out explicitly as a closed-form expression — for instance, if , then . 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 by interchanging the roles of the dependent and independent variables. Example: The product log is an implicit function giving the solution for of the equation . Algebraic functions An algebraic function is a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable gives a solution for of an equation where the coefficients are polynomial functions of . This algebraic function can be written as the right side of the solution equation . Written like this, is a 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: Solving for gives an explicit solution: But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as , where is the multi-valued implicit function. While explicit solutions can be found for equations that are quadratic,