Example OAF.CPPF.4 : $P(x)=\frac 2 {(x−1)^3}+2$.
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.