Inverse of $y = x^3 + x $? Can you help me find the inverse function for $y = x^3 + x$?  This question was posed at the beginning of AP Calculus, so we can't use any math beyond precalc.

Thanks!
Write $x^{3}+x-y=0$ and make $x=u+v$. You get $$u^{3}+v^{3}-y+\left(
u+v\right) \left( 3uv+1\right) =0.$$ Then solve the system

$$\left\{ 
\begin{array}{c}
u^{3}+v^{3}=y \\ 
u^{3}v^{3}=-\frac{1}{27},
\end{array}
\right. $$

which is equivalent to solving a quadratic equation in $u^3$ or $v^3$, because you know the sum of the two numbers $u^3,v^3$ and their product, e.g. the equation 

$$\left( u^{3}\right) ^{2}-yu^{3}-\frac{1}{27}=0.$$

This technique is known as Cardano's method.

Reference: Sebastião e Silva and Silva Paulo, Compêndio de Álgebra, VII ano, pp.215-216, 1963
Its inverse function is $ x= y^3-y $

It is obtained by interchanging $x$ and $y$ or by mirroring graph with respect to line through origin $ x=y. $