Question

If f(x) = x/(2 - x) then, find f^-1 (- 2) ​

Answer 1

We should confirm that f(x) has an inverse in the first place. If it does, then

`f\left(f^(-1)(x)\right) = x`

Given that f(x) = x/(2 - x), we have

`f\left(f^(-1)(x)\right) = (f^(-1)(x))/(2 - f^(-1)(x)) = x`

Solve for the inverse:

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

`f^(-1)(x) = 2x - x f^(-1)(x)`

`f^(-1)(x) + x f^(-1)(x) = 2x`

`f^(-1)(x) (1 + x) = 2x`

`f^(-1)(x) = (2x)/(1+x)`

Then

`f^(-1)(-2) = (2(-2))/(1-2) = \boxed{4}`

Note that this is the same as solve for x when f(x) = -2 :

x/(2 - x) = -2 ⇒ x = 4