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Write the domain for the given rational function:

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?

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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.

Write the domain for the given rational function: f(x) = (x^2 - x - 12) / (x^2 - 5x - 14)

Learn how to determine the domain of the rational function f(x) = (x^2 - x - 12) / (x^2 - 5x - 14). This problem explores key algebraic concepts, including factoring quadratic expressions and applying the domain exclusion principle. Suitable for high school students in Grades 9-12.

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