In our calculus textbook, it claims without proof that the polynomial function $f(x) = x^3+x$, which is a bijection of the real line with itself, has no closed-form inverse function.
What would be two odd degree, degree 5 or higher monic polynomials $p(x) \ne q(x)$ with rational coefficients with everywhere positive derivative, Galois group that is not solvable (so they are truly bijections of the real line with no elementary inverse functions), and which do not commute?
If such polynomials exist and forming the 1-manifold $C$ by $p(x) = q(y)$ in the plane, what would be an elementary parametrization of $C$ or some subarc thereof? Is there some canonical way to parametrize $C$? Just choosing $\gamma: [a,b] \to \mathbb{R}$ with $\gamma(t) = (q(t),p(t))$ won't work, because $p$ and $q$ don't commute, nor will choosing $\gamma: [a,b] \to \mathbb{R}$ with $\gamma(t) = (t,(q^{-1} \circ p)(t))$ or $\gamma(t) = ((p^{-1} \circ q)(t),t)$, as $p^{-1}$ and $q^{-1}$ are non-elementary.