Draw a mapping diagram and graph of $P$ yourself, or consider the GeoGebra figures and table below.

Martin Flashman, Nov. 7, 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, $P(x)=\frac 2 {(x−1)^3}+2$. 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.

The number $1$ is not in the domain of $P$, so the arrow from $x=1$ does not connect to a point in the target. This visualizing the "vertical asymptote" at $x=1$ on the graph of $P$.

There is the unique X- intercept as well. $P(a)=0$ when $\frac 2 {(a−1)^3}+2=0$ or $(a-1)^3 = -1$, so $a-1 = -1$ and $a=0$.

The function is decreasing for all $x \in (-\infty,1)$ and $x \in (1,\infty)$.

The function has odd symmetry with respect to the point $(1,2)$.

When the value of $x$ is very large, the values of $f(x)$ are very close to $2$ on the target axis. This visualizes that the line $y=2$" is a horizontal asymptote on the graph of $P$.

You can check this by moving $x$ on the GeoGebra mapping diagram.