What is the inverse function for #f(x) = x/(x-2)#?

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Answer 1 · 2015-09-22T23:09:53

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

Setting #y = f(x)#

#y = x/(x-2)#

this may be rearranged (intermediate steps shown) as follows

#y(x-2) = x#

#x*y - 2 y = x#

#x*y - x = 2 y#

#x(y - 1) = 2 y#

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

This expression shows #x# in terms of #y#.

That is, it is the inverse of function #f(x)#.

That is,

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

as required.