Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).
$f\left(x\right)=\frac{x}{x-2}$, for $x\gt{2}$

Verified step by step guidance

1. To find the inverse of the function \( f(x) = \frac{x}{x-2} \), start by replacing \( f(x) \) with \( y \). So, we have \( y = \frac{x}{x-2} \).
2. Next, solve for \( x \) in terms of \( y \). Begin by multiplying both sides by \( x-2 \) to eliminate the fraction: \( y(x-2) = x \).
3. Distribute \( y \) on the left side: \( yx - 2y = x \).
4. Rearrange the equation to isolate terms involving \( x \) on one side: \( yx - x = 2y \).
5. Factor out \( x \) from the left side: \( x(y - 1) = 2y \). Finally, solve for \( x \) by dividing both sides by \( y - 1 \): \( x = \frac{2y}{y - 1} \). This expression represents \( f^{-1}(x) \) when you replace \( y \) with \( x \).