Question

Find the inverse of $f\left(x\right)=x^3+x$

Answer 1

$f\left(x\right)=x^3+x$
$y=x^3+x$
$\Rightarrow x^3+x-y=0$
To solve the above equation we cand use method of niccolo
$x=u-v$
where $u={[\frac{y}{2}+{(\frac{y^2}{4}+\frac{1}{27})}^{\frac{1}{2}}]}^{\frac{1}{3}}$
$v={[\frac{y}{2}+{(\frac{y^2}{4}+\frac{1}{27})}^{\frac{1}{2}}]}^{\frac{1}{3}}$
$x={[\frac{y}{2}+{(\frac{y^2}{4}+\frac{1}{27})}^{\frac{1}{2}}]}^{\frac{1}{3}}-{[\frac{-y}{2}+{(\frac{y^2}{4}+\frac{1}{27})}^{\frac{1}{2}}]}^{\frac{1}{3}}$
Now we will replace y with x
$f^{-1}\left(x\right)={[\frac{x}{2}+{(\frac{x^2}{2}+\frac{1}{27})}^{\frac{1}{2}}]}^{\frac{1}{3}}-{[\frac{-x}{2}+{(\frac{x^2}{2}+\frac{1}{27})}^{\frac{1}{2}}]}^{\frac{1}{3}}$