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.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