<s>[INST] Inverse function of a polynomial $x(x-1)(x+1) = 0 $ How to get the interval to get the inverse function of the polynomial $$x(x-1)(x+1) = f(x) $$, to show that the function is surjective.
The interval where the function is surjective is $x > 1$ and $x < -1$, how to get inverse on this interval ?
 [/INST] Consider $$x(x-1)(x+1) =x^3-x= y$$ and then th cubic equation
$$x^3-x-y=0$$ for which the discriminant $\Delta=4-27y^2$ must be negative in order to have only one real root. This defines
$$-\frac{2}{3 \sqrt{3}} <y <\frac{2}{3 \sqrt{3}}$$
Using the hyperboic method for only one real root, we then have
$$x=\frac{2}{\sqrt{3} }\frac{ |y|}y   \cosh \left(\frac{1}{3} \cosh ^{-1}\left(\frac{3 \sqrt{3}
   }{2}|y|\right)\right)$$
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