How do you find the inverse of #1-ln(x-2)=f(x)#?

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Answer 1 · 2016-01-14T03:37:31

Inverse x and y.

#f^-1(x)=e^(1-x)+2#

The least formal way, (but easier in my opinion) is replacing x and y, where #y=f(x)#. Therefore, the function:

#f(x)=1-ln(x-2)#

#y=1-ln(x-2)#

Has an inverse function of:

#x=1-ln(y-2)#

Now solve for y:

#ln(y-2)=1-x#

#ln(y-2)=lne^(1-x)#

Logarithmic function #ln# is 1-1 for any #x>0#

#y-2=e^(1-x)#

#y=e^(1-x)+2#

Which gives the inverse function:

#f^-1(x)=e^(1-x)+2#