Implicit and explicit equations

An implicit equation is a relation of the form R(x₁, …, x_n) = 0, where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x² + y² − 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 x² + y² − 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.

Examples

Inverse functions

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

$y=g(x)$

for x in terms of y. This solution can then be written as

$x=g^{-1}(y)\,.$

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) = Template:Sfrac(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.

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 x gives a solution for y of an equation

$a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0\,,$

where the coefficients a_i(x) are polynomial functions of x. This algebraic function can be written as the right side of the solution equation y = f(x). Written like this, f 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:

$x^{2}+y^{2}-1=0\,.$

Solving for y gives an explicit solution:

$y=\pm {\sqrt {1-x^{2}}}\,.$

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation as y = f(x), where f is the multi-valued implicit function.

While explicit solutions can be found for equations that are quadratic, cubic, and quartic in y, the same is not in general true for quintic and higher degree equations, such as

$y^{5}+2y^{4}-7y^{3}+3y^{2}-6y-x=0\,.$

Nevertheless, one can still refer to the implicit solution y = f(x) involving the multi-valued implicit function f.

Resources

• Implicit Function, Wikipedia
• Differentiating Implicit and Explicit Functions, Math Mutts
• Implicit Differentiation, Openstax