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.

1 Answer
Shwetank Mauria
Nov 29, 2017

#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]}

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