Approximation for fractional function

Question

Let $x$ be a non-negative variable. Is there any good exponential approximation for the following fractional function, \begin{align} f(x)=\frac{x}{ax+b} \end{align} where $a$ and $b$ are non-negative real numbers. Is there any other kind of good approximations except exponential functions?

Answer 1

Your function $\frac{x}{ax+b}=\frac{1}{a}-\frac{b}{a(ax+b)}$ is a hyperbola. It has a vertical and horizontal asymptote whereas the function you want to approximate it by has only a horizontal asymptote. This is going to result in some part of the graph being a terrible approximation. So unless you can specify some range then the approximation will always be bad.

I would recommend trying for the same horizontal asymptote but beyond that it will never be a good fit.

Edit: Assuming you are interested in only $x\ge0$ part of the graph and $a$ and $b$ are both positive then there is a y-intercept of $0$. So a graph of $\frac{1}{a}(1-c^{-x})$ would give some similarity. Adjusting the value of $c$ would let you tweak the fit.