Inverse of $y=\frac{1}{27}(x^5+2x^3)$? Is it possible to find the inverse of a function such as $y=\frac{1}{27}(x^5+2x^3)$? This question was posed to me by an AP Calculus AB student. The function seems to be $1$-to-$1$ and have an inverse. However, I don't know a way to solve it. Symbolab doesn't seem to know either. The student was recalling the question from memory so I'm not $100\%$ sure this was the exact question. He also mentioned that a piece of information, $a=7$, might be a part of it. He was also under the impression that the question had to do with derivatives. Thoughts?

While the function is indeed one-to-one, it is impossible in general to find an analytic expression for $x$ even at $y=1$ – the polynomial that has to be solved in that case, $x^5+2x^3-27$, has Galois group $S_5$.
Nevertheless, since the function is monotone and odd, Newton's method works very well to obtain a numerical result.
