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

1 Answer
Alan P.
Nov 26, 2015

If f^(-1)(x) is the inverse of f(x)=(e^x)+1 then
color(white)("XXX")f^(-1)(x)=ln(x-1)
(with some obvious limitations since ln(x-1) is not defined for x<=1)

Explanation:

By definition of inverse
color(white)("XXX")f(f^(-1)(x)) = x

and since f(x)=(e^x)+1

color(white)("XXX")f(f^-1)(x)) = e^(f^(-1)(x))+1

and therfore
color(white)("XXX")e^(f^(-1)(x))+1 = x

color(white)("XXX")e^(f^(-1)(x) = x-1

Taking the natural log of both sides:
color(white)("XXX")f^(-1)(x) = ln(x-1)

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