Draw a mapping diagram and graph of $P$ yourself, 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, $P(x)=\frac 1 2(x−1)^3+2.$. 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.

When the point in the domain is $0$, the arrow points to $P(1) = 0+ 4 = 4$ visualizing the "Y-intercept" on the graph of $P$.

There is the unique X- intercept as well. $P(a)=0$ when $\frac 1 2(a−1)^3+4=0$ or $(a-1)^3 = -8$, so $a-1 = -2$ and $a=-1$.

The function is increasing for all $x \in (-\infty,\infty)$.

The function has odd symmetry with respect to the point $(1,4)$.

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