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) = \frac1{f^(-1)(x)+2} = x`

Solve for
`f^(-1)(x)` :

`\frac1{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) = \frac{1-2x}x \\\\ \boxed{f^(-1)(x) = \frac1x - 2}`

(provided that x ≠ 0 and x ≠ -2)