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.

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