This example presents a (piecewise) case defined function $f$, implicitly defined with respect to the equation $G(x,y) = x^2+y^2−4=0$ with a table of data, a graph and a mapping diagram.

Notice how the arrows on the mapping diagram are paired with the
points on the graph of the equation $G(x,y) = x^2+y^2−4=0$.

You can move the point for $x$ on the mapping diagram to see how
the function value changes both on the diagram and on the graph.

Notice the discontinuity of the function on the mapping diagram is
evidenced by the sudden jump in the arrow as the value of $x$
passes through $0$.

Check the upper box to show the graph with points on the graph and
arrows on the diagram to match the data in the table.

Once visible after checking the upper box you can change the
slider for $n$, a number that controls the number of points and
arrows shown. $n$ ranges from $1$ to $10$.

Check the lower box to show the confirmation that the paired data
do satisfy the equation: $G(x,y) = x^2+y^2−4=0$.

This confirmation also appears in the third (hidden) column of the
spreadsheet.

Martin Flashman, Nov. 9, 2013. Created with GeoGebra