Interesting problem!
Finding inverse of #y=e^x/(1+2e^x)#
For inverse understand that the graph would be a reflection over the #x=y# line. So the starting point would be to understand #(x,y)->(y,x)# when finding the inverse
Step 1: Swap #x# and #y#
#x = e^y/(1+2e^y)#
Step 2: Solve for #y#
We would start by multiplying both sides with #(1+2e^y)#. This is done to remove the denominator.
#x(1+2e^y) = e^y#
Use distributive property.
#x+2xe^y=e^y#
Collect all terms containing #e^y# to one side of the equation. Subtracting both sides by #2xe^y# should do the trick.
#x = e^y-2xe^y#
Factor out #e^y# from the right side of the equation. This is the reverse process of distribution.
#x = e^y(1-2x)#
We are solving for #y# and for that we need #e^y# isolated. To remove #(1-2x)# we would divide both sides by #(1-2x)#.
#x/(1-2x) = e^y#
To solve for #y# we need to convert this equation to natural logarithmic equation.
Take #ln# on both sides.
Note: #ln(e^a) = a#
#ln(x/(1-2x)) = y#
#y=ln(x/(1-2x))# This is the equation of the inverse.
