The given function is:

$f(x) = \frac{x - 3}{x^2 - x - 6}$

To find the domain of this function, we need to determine where the denominator is not equal to zero, as division by zero is undefined.

Step 1: Solve for the values that make the denominator equal to zero.

Set the denominator equal to zero:

$x^2 - x - 6 = 0$

Step 2: Factor the quadratic equation.

$x^2 - x - 6 = (x - 3)(x + 2)$

Step 3: Solve for the roots.

Set each factor equal to zero:

$x - 3 = 0 \quad \text{or} \quad x + 2 = 0$

Thus,

$x = 3 \quad \text{or} \quad x = -2$

Step 4: Exclude these values from the domain.

The function is undefined when $x = 3$ or $x = -2$, so the domain is all real numbers except these two values.

Final Answer:

The domain of $f(x)$ is:

$\boxed{(-\infty, -2) \cup (-2, 3) \cup (3, \infty)}$

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Would you like more details on how to find the domain? Or do you have other questions on related concepts? Here are some questions that could expand on this:

1. How do you identify vertical asymptotes from the domain?
2. Can the function be simplified before determining the domain?
3. What happens if the numerator equals zero?
4. How would you find the domain of a rational function with higher-degree polynomials in the numerator and denominator?
5. How do you analyze the behavior of the function at the values where the denominator equals zero?

Tip: When working with rational functions, always factor the denominator completely to identify the values that make it zero, as these are the restrictions on the domain.