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.