What is the inverse of the function f(x)=x^3-3x for different intervals?

[Ackbach]

Here is this week's POTW:

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Let $f:\mathbb{R}\to\mathbb{R}$ be the function given by $f(x)=x^3-3x$. Calculate $f^{-1}([-2,2]), f^{-1}((2,18)), f^{-1}([2,18)),$ and $f^{-1}([0,2])$.

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[Ackbach]

No one answered this week's POTW correctly. Here is my solution:

Spoiler

A plot of the function follows using Desmos. You can adjust it so that it captures all $y$-values from $-2$ to $18$.
[desmos="-10,10,-10,10"]x^3-3*x[/desmos]
From this graph, we can see by inspection that
\begin{align*}
f^{-1}([-2,2])&=[-2,2] \\
f^{-1}((2,18))&=(2,3) \\
f^{-1}([2,18))&=\{-1\}\cup [2,3) \\
f^{-1}([0,2])&=[-\sqrt{3},0]\cup[\sqrt{3},2].
\end{align*}