How to find the inverse function involving the exponential function?

Question

Given: $f(x)= \dfrac{e^x}{1+9e^x}$ ,

what steps would I take to find its inverse? I tried following the steps on finding the inverse of a normal function but I keep getting one of the variables to cancel out.

Answer 1

We want to solve the equation $$y=\frac{e^x}{1+9e^x}$$ for $x$ as function of $y$. $$(1+9e^x)y=e^x$$ $$y+9e^xy=e^x$$ $$y=e^x-9e^xy$$ $$y=e^x(1-9y)$$ $$\frac y{1-9y}=e^x$$ $$e^x=\frac y{1-9y}$$ $$\ln(e^x)=\ln(\frac y{1-9y})$$ $$x=\ln(\frac y{1-9y})$$