An equation that relates two variables can often be used to define one or more functions that connect the variables with the equation.
This is what is described by saying a function is defined implicitly by an equation.

An illustration of this general statement is the connection of inverse functions examined in OW.IVPPF.

Thus by examining the equation $y^2 -x =0$ we come to define two functions, $ f_+(x) =\sqrt x$ and $f_-(x)=-\sqrt x$, both of which can be substituted in the equation for $y$ to make the original equation true.
 [In fact, by using definition by cases, there are infinitely many distinct functions that can satisfy the equation]

Example OW.0 also illustrates the general way in which a function  can be defined implicitly by an equation.


Summary: A function $f$ is defined implicitly by an equation $G(x,y)=0$ if for any $x$ in the domain of $f$, $G(x, f(x)) = 0$.
An implicit function for an equation is not necessarily unique. Thus, visualizing such a function with a graph or mapping diagram can be verified visually by connecting the function to a visualization of the defining equation with a graph or diagram.

The following examples show some of the ways implicit functions can be visualized with mapping diagrams. 

Example OW.IMPL.1 : $G(x,y) = x^2 - y^2 = 0$.

Example OW.IMPL.2 :  $G(x,y) = x^3 - y^2 = 0$.

Example OW.IMPL.3 : $G(x,y) = x^2 - y^3 = 0$.

Example OW.DIMPL.0 : Dynamic Visualization of the Implicit Functions: Graphs and Mapping Diagrams