How do you find the inverse of y = x/(x+1) and is it a function?

1 Answer
May 23, 2016

x = y/(1-y), which is a function.

Explanation:

Isolate x as follows:

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

So, adding (1/(x+1) - y) to both ends we find:

1/(x+1) = 1-y

Taking reciprocals of both sides we get:

x+1 = 1/(1-y)

Subtracting 1 from both sides we get:

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

This uniquely determines x for any chosen value of y (except y=1 where it's not defined at all), and so is a function.

So if f(x) = x/(x+1) then f^(-1)(y) = y/(1-y)

  • The domain of f(x) is (-oo, -1) uu (-1, oo)

  • The domain of f^(-1)(y) is (-oo, 1) uu (1, oo)

So:

  • The range of f(x) is (-oo, 1) uu (1, oo)

  • The range of f^(-1)(y) is (-oo, -1) uu (-1, oo)