To evaluate $\left(\frac{f}{g}\right)(x)$ given that $f(x) = \sqrt{x-2}$ and $g(x) = \frac{1}{x}$, follow these steps:

Step 1: Write the expression for $\left(\frac{f}{g}\right)(x)$

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Step 2: Substitute the given functions $f(x)$ and $g(x)$ into the expression

$\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x-2}}{\frac{1}{x}}$

Step 3: Simplify the expression

Dividing by $\frac{1}{x}$ is equivalent to multiplying by $x$: $\left(\frac{f}{g}\right)(x) = \sqrt{x-2} \times x$

Step 4: Write the final answer

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

This is the expression for $\left(\frac{f}{g}\right)(x)$.

Would you like more details or have any questions? Here are some related problems you might find interesting:

1. Find $\left(\frac{f \cdot g}{h}\right)(x)$ if $f(x) = \sqrt{x-2}$, $g(x) = \frac{1}{x}$, and $h(x) = x^2$.
2. Evaluate $\left(f \cdot g\right)(x)$ for the same functions $f(x)$ and $g(x)$.
3. Determine the domain of $\left(\frac{f}{g}\right)(x)$.
4. Simplify $\left(\frac{f}{h}\right)(x)$ if $h(x) = x+2$ and $f(x) = \sqrt{x-2}$.
5. Find the inverse of the function $f(x) = x\sqrt{x-2}$.

Tip: When dividing by a function, always ensure that the denominator is not zero to avoid undefined values.