# Approximation of fractional function that has real-power numerator

I have the function

$f(x)=\frac{(1+x)^k}{1+ax}$, where $x>0, 0<a<k<1$.

The function has only one maximum at $x_0=\frac{a-k}{a(k-1)}$, increases on the left of $x_0$ and decreases on the right of $x_0$.

I want to find an approximation for $f(x)$ on the right of $x_0$. There are two constraints of the approximation:

1. It only needs to work for $f(x)>1$.

2. The approximation should have fairly simple (preferably unique) inverse function.

Any hint is appreciated.

• Could you explain the purpose of the approximation and/or provide criteria for determining how good it is? Otherwise there are far too many solutions to describe and they're all arbitrary. – whuber May 17 '18 at 20:25
• Actually I want to find a simple expression with respect to $x$ for $x_r=f^{-1}_r(f(x))$, where $0<x<x_0$ and $f_r^{-1}$ is the inverse function of $f(x)$ on the right of $x_0$. I intend to find an approximation for $f(x)$ on the left and an approximation for $f(x)$ on the right of $x_0$. I think the left can be approximated by a parabolic curve when $a$ is close to $k$. This question is for the right part. – Tri Nguyen May 17 '18 at 20:59