How do you find the inverse of f(x) = (x - 2) / (x + 2)?

Dec 13, 2015

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

Explanation:

$f \left(x\right) = \frac{x - 2}{x + 2}$

$y = \frac{x - 2}{x + 2}$

Switch $x$ for $y$ and $y$ for $x$

$x = \frac{y - 2}{y + 2}$

Multiply both side by $y + 2$

$\left(y + 2\right) \cdot x = \left(\frac{y - 2}{y + 2}\right) \left(y + 2\right)$

Distribute

$x y + 2 x = y - 2$

Bring all "y" term to one side and factor

$2 x + 2 = y - x y$

Factor, and solve for $y$

$2 \left(x + 1\right) = y \left(1 - x\right)$

$\frac{2 \left(x + 1\right)}{1 - x} = y$

Replace $y$ with inverse notation ${f}^{- 1} \left(x\right)$

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