Implicit function

Throughout grade school mathematics, equations usually define single functions. Take for example, the equation $y=3x+2$ which defines $y$ as a functionAlternatively, $y$ might be described as a dependent variable of the independent variable $x$. of $x$. Equations of this phenotype are called explicit functions.[cite:@openstax2020calculus]

But, of course, not all equations define $y$ explicitly. The equation $y-x^2=2$ is one such relation that defines the implicit function $y=x^2+2$. Further, not all equations define just one function. The equation defining the unit circle, $x^2+y^2=1$ is an obvious non-function,Plotting the equation reveals its failure of the "vertical line test" — there are values of $x$ with multiple corresponding $y$-values, disqualifying it as a function $ℝ\to ℝ$, but not as a multifunction or set-valued function }:). but it does define the implicit functions $y=\sqrt{1-x^2}$ and $y=-\sqrt{1-x^2}$. The key idea is that, while an equation may not correspond to a function as a whole, it may correspond to various functions each with certain conditions.

Definition

Formally, an implicit equation is a relation of the form $R(x_1,\cdots ,x_n)=0$, where $R$ is a function of $n$ variables.[cite:@enwiki:1286472469] An implicit function is any corollary function $f$ that defines one of the variables $x_k$ in terms of the rest.