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

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Answer 1 · 2015-11-21T18:40:19

Let #y = f(x)# and solve for #x# to find:

#f^(-1)(y) = (y-1)/(y-2)#

#y = f(x) = (2x-1)/(x-1) = (2x-2+1)/(x-1) = (2(x-1)+1)/(x-1)#

#= 2+1/(x-1)#

Subtract #2# from both ends to get:

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

Hence:

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

Add #1# to both sides to get:

#x = 1/(y-2)+1 = (1+(y-2))/(y-2) = (y-1)/(y-2)#

So #f^(-1)(y) = (y-1)/(y-2)#