Example OAF.CNPF.2 : $P_{-4}(x)=x^{-4} = \frac 1 {x^4}$.
Draw a mapping diagram and graph of $P_{-4}$ yourself, or consider the GeoGebra figures and table below.
Martin Flashman, Nov. 1, 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,
$P_{-4}(x)=\frac 1 {x^4}$. which corresponds to the core function's
value for $x$.
Check the box in Geogebra to see more points and arrows
corresponding to data on the table.
The number $x = 0$ is not in the domain, so there is no
value on the target axis for the arrow from $x=0$ in the source axis
to hit. This visualizes the "Y-axis" as a pole on the graph of
$P_{-4}$.
When the value of $x$ is very large, the values of $f(x)$ are
very close to $0$ on the target axis. This visualizes that the
"X-axis" is a horizontal asymptote on the graph of $P_{-4}$.
The function is increasing
for all $x \in (-\infty,0)$ and decreasing for $x \in
(0,\infty)$.
The function has even
symmetry.
You can check this by moving $x$ on the GeoGebra mapping diagram.