For example, $P_{-3}(x) = x^{-3} = \frac 1 {x^3}$.

**A core negative power function is the simplest of rational
functions . **

Since "division by $0$" doesn't make sense, $0$ is not in
the domain of $P_n$ when $n<0$, the line $x=0$ has no point in
common with the graph of $P_n$ and $x=0$ is a "pole" for $P_n$.

A core negative power function can understood as the composition
of a core positive power function with the core negative power
function $P_{-1}$.

For example $P_{-5} = P_{-1} \circ P_5$.

Simple modifications of $P_n$ can involve compositions with the
core linear functions, giving functions in a "vertex" form similar
to that for quadratic
polynomials:

When $n$ is an even number

When $x$ is a very large positive or negative number, then $f(x)$ is very close to $k$, which is described on the graph by saying $y = k$ is a horizontal asymptote for the graph of $f$.

We explore how mapping diagrams for core negative power functions help us visualize and understand further the meaning of the shape of the graph of $P_n$ when $n<0$..