Okay, here we go - tough with the limitations of this screen, but we'll go for it!
Like finding all inverse functions, first exchange x and y in the problem. Then, re-solve the problem to isolate y again. Then, replace y with F^-1(x). y=e^x/(1+2e^x) [starting equation] x=e^y/(1+2e^y) [swap x and y] x(1+2e^y)=e^y [multiply both sides by (1+2e^y) x+2xe^y=e^y [distribute x] x=e^y-2xe^y [subtract 2xe^y from both sides]
x= e^y(1-2x) [factor out e^y] x/(1-2x) = e^y [divide both sides by (1-2x)] ln(x/(1-2x)) = ln e^y [take ln of both sides]
ln (x/(1-2x)) = y [ln and e are inverses]
f^-1(x)= ln (x/(1-2x))
Go to desmos.com or use your TI graphing calculator to graph both the function and it's inverse to visually check that they are symetrical across the line y=x. You're all set.
The difference with this problem from others is the desire to quickly take the ln of all e terms. You need to wait until you have the variables in the right position before doing that.
