Are all polynomial functions invertible if we allow non-algebraic functions in the inverse?

I remember reading about somr theorem which proves that polynomials with a degree higher than 4 can't be solved algebraically with the help of radicals. And, I've looked algebraic formulas for solving equations of degree 3 and 4. They both were very huge and impractical unlike the quadratic formula. However, if we allow trigonometric, logarithmic and exponential functions in the inverse, then is it possible to get a formula of x in terms of the coefficients for polynomials of degree higher than 4?

• I cannot prove it, but I don't think that such functions can help much. If you want to find a root of high-degree polynomial with large precision, just use Newton method. – Wolfram Jan 29 '17 at 10:06
• The title is unlucky. You want to solve $p(x)=0$, where $p$ is a polynomial. To verify, whether $p$ is invertible, you search a function $q(x)$, such that $p(q(x))=q(p(x))=x$ for all $x$ – Peter Jan 29 '17 at 10:06
• @Wolfram Newton-method works in most cases, but not always. But in general, it is a very good choice. – Peter Jan 29 '17 at 10:12