A core positive power function has the form $P_n(x) = x^n$ where $n > 0$ is a natural number.
For example: $P_3(x) = x^3$.

When $n$ is an odd integer, $P_n$ is an increasing function and therefore is also $1:1$, with a unique inverse function $InvP_n = P_n^{-1}$ where $(InvP_n \circ P_n) (x) = (P_n^{-1} \circ P_n) (x)=x$  for all $x$.

In this case $InvP_n(x)=\sqrt[n]x$, the $n$th root of $x$.

When $n$ is an even integer, $P_n$ is a decreasing function for $(- \infty,0)$ and an increasing function for $(0,\infty)$.

The range of $P_n$ is $[0,\infty)$. Focusing on the interval $[0. \infty)$ we have $P_n$ is $1:1$, with a unique inverse function$InvP_n = P_n^{-1}$ where $(InvP_n \circ P_n) (x) = (P_n^{-1} \circ P_n) (x)=x$  for all $x \in [0,\infty)$.
In this case $P_n(x)=+\sqrt[n]x$, the nonnegative $n$th root of $x$.

We explore how mapping diagrams for the core positive polynomial functions help us visualize and understand further the meaning of the shape of the graph of $InvP_n$.

Example OW.ICPPF.1 : $InvP_2(x) = \sqrt[2] x$.

Example OW.ICPPF.2 $InvP_3(x) =\sqrt[3] x$.

Example OW.ICPPF.3 : $InvP_4(x) =\sqrt[4] x$.

Example OW.ICPPF.4 : If $P(x) =\frac 1 2 (x-1)^3 + 2$ then $InvP(x) = \sqrt[3] {2 (x - 2)}+1$.

Example OW.DICPPF.0 : Dynamic Visualization of the Inverse Core Positive Power Function: Graphs and Mapping Diagrams