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

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.