Question

How do we find the inverse of the function `y=e^(2x)/(7+e^(2x))`?

Answer 1

Inverse function of `f(x)=y=e^(2x)/(7+e^(2x))` is `y=1/2ln((7x)/(1-x))`

Explanation:

Let `f(x)=y=e^(2x)/(7+e^(2x))`

i.e. `(7+e^(2x))/e^(2x)=1/y` or

`7/e^(2x)+1=1/y` or

`7/e^(2x)=1/y-1=(1-y)/y` or

`e^(2x)=(7y)/(1-y)` or

`ln((7y)/(1-y))=2x` and hence

`x=1/2ln((7y)/(1-y))`

Hence inverse function of `f(x)=y=e^(2x)/(7+e^(2x))` is `y=1/2ln((7x)/(1-x))`