# Inverse function of $x + x^q$ with rational $q$

Consider the function:

$$f_{q}\left(x\right)=x+x^{q}$$

where $q\in\mathbb{Q},q>0$ and $x \in \mathbb{R}$, $x\geq 0$.

I am wondering what would be a method for inverting this function. It is monotone increasing on the non-negative reals as far as I see and so should be invertible.

Is there a closed form for the inverse function? Otherwise, could we express it as a series or an integral perhaps?

Thank you

• It'n not something that you can do in general. – Surb Aug 13 '17 at 12:00
• Of course you can do it in general. Puiseux series If $q=a/b$ with $(a,b)=1$ replace $x$ with $X^b$. The equation becomes $y=X^b+X^a$. Then you can expand $X$ as a Puiseux series. – uSir470888 Aug 13 '17 at 12:08
• See the introduction of ... cs.uwaterloo.ca/research/tr/1993/03/W.pdf – Donald Splutterwit Aug 13 '17 at 12:56
• For $q=a/b,a\le5,b\le5$, and $(a,b)=1$, the inverse can be explicitly written using regular radicals and Bring radicals. – Simply Beautiful Art Aug 13 '17 at 13:38