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

$f\left(f^{-1}(x)\right) = \dfrac{f^{-1}(x)}{2 - f^{-1}(x)} = x$

Solve for the inverse:

$\dfrac{f^{-1}(x)}{2 - f^{-1}(x)} = x$

$f^{-1}(x) = 2x - x f^{-1}(x)$

$f^{-1}(x) + x f^{-1}(x) = 2x$

$f^{-1}(x) (1 + x) = 2x$

$f^{-1}(x) = \dfrac{2x}{1+x}$

Then

$f^{-1}(-2) = \dfrac{2(-2)}{1-2} = \boxed{4}$

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

x/(2 - x) = -2   ⇒   x = 4