To determine the domain of the rational function:

$f(x) = \frac{x^2 - x - 12}{x^2 - 5x - 14}$

we need to identify the values of $x$ that make the denominator equal to zero, as these values will make the function undefined.

Step 1: Factor the Denominator

The denominator is: $x^2 - 5x - 14$

To factor this quadratic expression, we need two numbers that multiply to $-14$ and add up to $-5$. These numbers are $-7$ and $2$, so we can factor the denominator as: $x^2 - 5x - 14 = (x - 7)(x + 2)$

Step 2: Identify Values that Make the Denominator Zero

The values of $x$ that make the denominator zero are the solutions to: $(x - 7)(x + 2) = 0$

Solving each factor separately, we get: $x = 7 \quad \text{and} \quad x = -2$

Step 3: State the Domain

Since the function is undefined at $x = 7$ and $x = -2$, we exclude these values from the domain. Therefore, the domain of $f(x)$ is all real numbers except $x = 7$ and $x = -2$.

Final Answer

The domain of $f(x)$ is: $x \in \mathbb{R} \setminus \{7, -2\}$

Or in interval notation: $(-\infty, -2) \cup (-2, 7) \cup (7, \infty)$

Would you like a more detailed explanation of any step, or do you have any further questions?

Here are 5 related questions to explore further:

1. How would we find the domain if the function had a square root in the denominator?
2. What is the process for finding vertical asymptotes in rational functions like this one?
3. Can the function $f(x)$ be simplified further if the numerator also has $x - 7$ or $x + 2$ as factors?
4. How do we determine the range of a rational function?
5. What would the domain be if the denominator were a cubic function instead of a quadratic?

Tip: For any rational function, the domain is all real numbers except where the denominator equals zero. Always factor the denominator to find these points easily.