Draw a mapping diagram and graph of $P^{-1}$ yourself or consider the GeoGebra figure and table below.

Martin Flashman, Nov. 12, 2013, Created with GeoGebra

The GeoGebra figure illustrates the key property for this inverse: $(P^{-1} \circ P )(x) =x$ for all $ x$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 first target line at the point / number, $P(x)=\frac12(x−1)^3+2$ and from that point to $P^{-1} (x)=\sqrt[3]{2(x−2)}+1$.

When the point in the domain is $x =2$, the first arrow points to $P(2) = 1$ while the second arrow points to $P^{-1}(1)=2$.

Check the box in Geogebra to see more points and arrows corresponding to table for $P^{-1}$.

Both functions $P$ and $P^{-1}$ are increasing for all $x \in (-\infty,\infty)$.