Question

what is the inverse function of $f(x) = \frac{x}{x - 2}$? $f^{-1}(x) = \frac{2x}{x - 1}$; $f^{-1}(x) = \frac{x - 2}{x}$; $f^{-1}(x) = -\frac{2x}{x - 1}$; $f^{-1}(x) = \frac{2x}{x + 1}$

Explanation:

Step1: Set $y=f(x)$

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

Step2: Swap $x$ and $y$

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

Step3: Cross-multiply to isolate $y$

$x(y-2) = y$
$xy - 2x = y$

Step4: Group terms with $y$

$xy - y = 2x$
$y(x-1) = 2x$

Step5: Solve for $y$

$y = \frac{2x}{x-1}$

Answer:

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