Example OW.IMPL.3

Example OW.IMPL.3 : $G(x,y) = x^2 - y^3 = 0$.
Suppose $f(x) = x^{2/3} = (\sqrt [3] x )^2$.
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 $f$.

Note: Solving the equation for $y$ in terms of $x$ demonstrates that this implicit function is unique.

The function is decreasing for all $x$ in the intervals $(-\infty, 0]$ and increasing 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.