Example OW.ICPPF.4 : If $P(x)=\frac12(x−1)^3+2$ then
$P^{-1}(x)=\sqrt[3]{2(x−2)}+1$.
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 $x \in
(-\infty,\infty)$.