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Concept
Implicit function
Summary
In mathematics, an implicit equation is a relation of the form where R 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 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 − xex = 0.
Algebraic function
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
where the coefficients ai(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.
Official source
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