To solve the problem, let's go through the steps one by one.

Given:

• $f(x) = \frac{3}{x}$
• $g(x) = x - 2$

We are asked to find $\left( \frac{f}{g} \right)(x)$ and determine its domain.

Step 1: Find $\left( \frac{f}{g} \right)(x)$

The expression $\frac{f}{g}$ represents the function $\frac{f(x)}{g(x)}$.

So, $\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{\frac{3}{x}}{x - 2} = \frac{3}{x(x - 2)}$

Step 2: Determine the Domain

The domain of $\left( \frac{f}{g} \right)(x)$ is restricted by values of $x$ that would make the denominator zero.

1. $x \neq 0$, because $f(x) = \frac{3}{x}$ would be undefined.
2. $x \neq 2$, because $g(x) = x - 2$ would be zero.

Thus, the domain of $\left( \frac{f}{g} \right)(x)$ is: $(-\infty, 0) \cup (0, 2) \cup (2, \infty)$

Final Answer

• Function: $\left( \frac{f}{g} \right)(x) = \frac{3}{x(x - 2)}$
• Domain: $(-\infty, 0) \cup (0, 2) \cup (2, \infty)$

Would you like a further breakdown or details on any part of the solution?

1. How to evaluate the quotient of two functions?
2. What steps are necessary to determine the domain of a rational function?
3. Why do we exclude certain values from the domain?
4. How do you simplify complex fractions like this one?
5. What other examples involve combining functions with restricted domains?

Tip: When finding the domain of a function, always check for values that cause division by zero or negative square roots, as these restrict possible inputs.