Inverse of a Polynomial

If you have something like, say, x = 3*y^2, it's trivial to write y as a function of x (y = sqrt(x/3)).

But what about a polynomial? Like, say, x = 3*y^2 + 2*y. Is that doable "easily"?

You can usually only construct an inverse if the function is continuous and monotonic. It's possible to mess around with functions taking several values in some simpler cases (your example should be y = +/- sqrt(x/3)) but generally not feasible.