I was playing around with a graphing calculator, trying to find approximations for inverses of $f(x)=x^5+x+1$. This cannot be expressed with radicals or the like, but I wanted to see how close I could get with them. After some fooling around I found that if I replace $x=y$ we can get: $y^5+y+1=x$, then $y=(x-1-y)^{\frac15}$ call the LHS $g(x,y)$. Now, define: $y_{n+1}=g(x,y_n)$ and where $n$ goes through all natural numbers and $y_0$ is any constant.
If you try this yourself, you will see that $y_n \sim f^{-1}(x)$. I have thought about this for some time now and I have been unable to come up with an explanation. Why does this happen? And why are there intervals of x for which this approximation doesn't work?
2026-04-24 11:16:55.1777029415