What function $y=f(x,a)$ approximates the relation $1-y=(1-xy)^a$?
The domains of the variables are as follows: $x,y\in [0,1], a\in[1,\infty)$. Clearly $y=0$ is one solution, but I would like the other, non-trivial solution. So far I have tried taking the log of both sides and using a Taylor series, but this gave a (very) poor approximation for when $y$ is close to 1. I am unsure how to make these sorts of implicit equations into explicit functions, or even how to get a decent approximation. I'd appreciate either hints or answers, thanks!
You could at least perform one iteration of Newton method using $y_0=1$. This would give $$y_1=1-\frac{(1-x)^a}{1-a x (1-x)^{a-1}}$$ For example $$a=\pi \quad \text{and} \quad x=\frac 12\quad \implies \quad y_1=1-\frac{2^{-\pi }}{1-2^{-\pi } \pi }=0.824049$$ while the exact solution is $y=0.798037$