How do you find the inverse of #f(x) = x / (x + 8)#?

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Answer 1 · 2015-11-22T00:43:31

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

#f^(-1)(y) = (8y)/(1-y)#

#y = f(x) = x/(x+8) = (x+8-8)/(x+8) = 1-8/(x+8)#

Hence (adding #8/(x+8)-y# to both ends):

#8/(x+8) = 1 - y#

Hence (multiplying both sides by #(x+8)/(1-y)#):

#x+8 = 8/(1-y)#

So:

#x = 8/(1-y) - 8 = 8/(1-y) - (8-8y)/(1-y) = (8y)/(1-y)#

So:

#f^(-1)(y) = (8y)/(1-y)#