How to invert a polynomial function such as $f(x)=x^{\beta}\left((x-1)^6+1\right)$?

Question

I'm trying to replicate a simulation study in a paper. For that I would need the inverse of this function:

$f(x)=x^{\beta}\left((x-1)^6+1\right), x\in[0,1]$

Plugging this unto Maxima returns:

solve(y=x^(beta)*((x-1)^6+1),x);
   beta        beta + 3                                                                            
[x     = y - x        ]

which doesn't help me one bit. The fact that it's used in this paper leads me to think that the inverse of this function is probably well known. It's just not known to me.

Alternatively, is there a general numerical strategy to find the inverse of a function for a grid of values of x?

Answer 1

As noted in the comments, there is no closed form for the inverse. There are numerical techniques for finding very good approximations. One of the simplest and best is called Newton's Method - you will have no trouble finding it.