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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