How do you find the inverse of y = (e^x)/(1+2e^x)?

1 Answer
Jan 11, 2016

Step by step working is shown below.

Explanation:

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.