Question

Find f^(-1)for the function f(x)= (1)/(x+2)

Answer 1

If $f(x)$ and $f^{-1}(x)$ are inverses of each other, then we have

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

Since $f(x) = \frac1{x+2}$, we have

$f\left(f^{-1}(x)\right) = \dfrac1{f^{-1}(x)+2} = x$

Solve for $f^{-1}(x)$ :

$\dfrac1{f^{-1}(x)+2} = x \\\\ 1 = x\left(f^{-1}(x)+2\right) \\\\ 1 = x f^{-1}(x) + 2x \\\\ x f^{-1}(x) = 1 - 2x \\\\ f^{-1}(x) = \dfrac{1-2x}x \\\\ \boxed{f^{-1}(x) = \dfrac1x - 2}$

(provided that x ≠ 0 and x ≠ -2)