What is an Implicit Function?
In mathematics, an implicit equation is a relation of the form {\displaystyle R(x_{1},\ldots ,x_{n})=0} {\displaystyle R(x_{1},\ldots ,x_{n})=0}, where {\displaystyle R} 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} {\displaystyle x^{2}+y^{2}-1=0}.
An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). Thus, an implicit function for {\displaystyle y} y in the context of the unit circle is defined implicitly by {\displaystyle x^{2}+f(x)^{2}-1=0} {\displaystyle x^{2}+f(x)^{2}-1=0}. This implicit equation defines {\displaystyle f} f as a function of {\displaystyle x} x only if {\displaystyle -1\leq x\leq 1} {\displaystyle -1\leq x\leq 1} and one considers only non-negative (or non-positive) values for the values of the function.1
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