# Implicit Function **A [[function]] defined by an implicit equation, which relates the value of the function with its arguments.** An *implicit equation* is of the form $f(x_{1},\dots,x_{n})=0$, where $f$ is a *multivariable function*, that relates several variables. Alternatively, it is an equation where all variables are on the *same side* of the equals sign. An *implicit function* is a function defined by an implicit equation, where one variable is considered the *value* of the function whilst the others are considered its *arguments*. >[!Explanation]- > For example, if there was an implicit equation of *3 variables*, $F(x,y,z)=0$, it can be used to implicitly define a function $z=f(x,y)$. > > $z$ would be considered the *value* of the function and $x$ and $y$ as its *arguments*, even though the equation *does not explicitly* express the relationship in that form. Often, more than one implicit function can be defined using the same implicit equation by *restricting the domain or codomain* of the function. Implicitly defined functions can be differentiated using [[implicit differentiation]]. >[!Example]+ > The *unit circle* can be described by the implicit equation > $x^{2}+y^{2}-1=0$ > If the values of $x$ are restricted to the interval $[-1,1]$ and $y$ is restricted to either *negative* or *non-negative* values, then the equation also defines $y$ as an *implicit function* of $x$. [^1] > > In this way, $y$ can then be considered the *value* of the function as in the typical sense of a *single variable* function, i.e. $F(x,y)=0$ and $y=f(x)$. > > ![[ImplicitFunctionExample.svg]] > If the implicit equation is rearranged for $y$, the two explicit forms of the functions can be derived. > $y=\pm\sqrt{1-x^{2}}$ [^1]: This is not the only method of restricting the domain and codomain to yield valid functions.