I derived a function in the form: $$ f(x)={\left[ 1 - \frac{N-m \choose x}{ N \choose x } \right]}^{\frac{N}{x}} $$ $m$ can be treated as a constant and $m=\alpha N$, where $ 0 < \alpha < 1$. $N$ also ban be regarded as a constant. I want to analyse how $f(x)$ changes with respect to $x$, i.e. how to minimize $f(x)$ with respect to $x$. I guess there could be a closed-form expression for $f(x)$. But I cannot figure it out. Therefore my question is:
Is it possible to derive a closed-form expression for $f(x)$? If not, is there any approximation method to analyse how $f(x)$ changes in terms of $x$.
Thank you!