Example OW.IMPL.1 : $G(x,y) = x^2 - y^2 = 0$.
Suppose $f(x) = x$ if $x \le 0$ and $f(x) = -x$ otherwise.
Draw a mapping diagram and graph of $f$ yourself, or  consider the GeoGebra figures and table below.

Martin Flashman, Dec. 21, 2013, Created with GeoGebra

Notice how the points on the graph are paired with the points on the mapping diagram.
Given a point / number, $x$, on the source line, there is a unique arrow  meeting the target line at the point / number, $f(x)$. which corresponds to the function's value for $x$
Check the box in Geogebra to see more points and arrows corresponding to data on the table.
Use the slider for $n$ to change the number of data points.
When the point in the domain is $0$, the arrow points to $f (0)= 0$ visualizing the "Y-intercept" on the graph of $f$.  Notice also the "hole"  on the graph at $(0,1)$ which is the "Y-intercept" for the linear function $g(x) = 2x+1$.

The function is increasing for all $x$ in the intervals $(-\infty, 0]$ and  decreasing for all $x$ in $(0,\infty)$.
You can check this by moving $x$ on the GeoGebra mapping diagram.
You can change the function to explore other implicit functions further.