What's the inverse of f(x)= e^x/(1+2e^x)?

[nfxgosu]

What's the inverse of f(x)= e^x/(1+2e^x)?

 

[Marin]

Hi, nfxgosu!

You just have to solve for 'x':

$$f(x)=y=\frac{e^x}{1+2e^x}$$

$$(1+2e^x)y=e^x$$

$$y+2ye^x-e^x=0$$

$$e^x(2y-1)=-y$$

$$e^x=-\frac{y}{2y-1} /ln$$

$$x=\ln\frac{y}{1-2y}$$

So, for the inverse function we have:

$$\overline{f}(x)=\ln\frac{x}{1-2x}$$

Best wishes, Marine