Question

Help with finding the inverse of a function?

Find the inverse of f(x)=ln(1-2x). Find the domain of f and the domain of f^-1. Justify your conclusions?

I'm not sure how to solve this problem, I think your supposed to put an e on both sides and bring up the exponents somehow? Also what does it mean by finding the domain of f? I think f^-1 just means the answer but I'm not sure about the domain of f.

Answer 1

`f^(-1)(x)=(1-e^x)/2`

Explanation:

For finding inverse of a function `y=f(x)`, find value of `x` in terms of `y`. Let this be `F(y)` and then `(F(x)` is the inverse function of `f(x)` and is written as `f^(-1)(x)`.

Here `y=f(x)=ln(1-2x)`

hence `1-2x=e^y`

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

hence `f^(-1)(x)=(1-e^x)/2`

Further, these functions are reflected across `y=x`, however while domain of `f(x)` is `(oo,1/2)`, domain of `f^(-1)(x)` is `(-oo,oo)` and hence there are certain limitations, as may be seen from their graphs below.

graph{(y-ln(1-2x))(2y-1+e^x)(x-y)=0 [-10.625, 9.375, -5.92, 4.08]}