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To find the domain of the given functions \( f(x) \) and \( g(x) \), we need to determine when the denominators of each function are non-zero, as division by zero is undefined.

1. **Domain of \( f(x) = \frac{x - 7}{x^2 + 14x + 49} \):**
   The domain is restricted by the denominator. We solve:
   \[
   x^2 + 14x + 49 = 0
   \]
   This can be factored as:
   \[
   (x + 7)^2 = 0
   \]
   So, \( x + 7 = 0 \) gives \( x = -7 \).

   Therefore, the domain of \( f(x) \) is all real numbers except \( x = -7 \). In interval notation:
   \[
   (-\infty, -7) \cup (-7, \infty)
   \]

2. **Domain of \( g(x) = \frac{x - 2}{x^2 - x - 2} \):**
   The domain is restricted by the denominator. We solve:
   \[
   x^2 - x - 2 = 0
   \]
   Factor the quadratic:
   \[
   (x - 2)(x + 1) = 0
   \]
   So, \( x - 2 = 0 \) gives \( x = 2 \) and \( x + 1 = 0 \) gives \( x = -1 \).

   Therefore, the domain of \( g(x) \) is all real numbers except \( x = -1 \) and \( x = 2 \). In interval notation:
   \[
   (-\infty, -1) \cup (-1, 2) \cup (2, \infty)
   \]

### Summary:
- **Domain of \( f(x) \):** \( (-\infty, -7) \cup (-7, \infty) \)
- **Domain of \( g(x) \):** \( (-\infty, -1) \cup (-1, 2) \cup (2, \infty) \)

Do you have any questions or need further details?

Here are 5 related questions you might explore:
1. What is the range of \( f(x) \) and \( g(x) \)?
2. How would the graph of each function behave at the points where the denominators are zero?
3. Can you find the vertical asymptotes of \( f(x) \) and \( g(x) \)?
4. What are the intercepts of the functions \( f(x) \) and \( g(x) \)?
5. How would the functions change if the numerators were different?

**Tip:** Always check the denominator for restrictions when determining the domain of rational functions.

The functions f(x) = (x - 7)/(x^2 + 14x + 49) and g(x) = (x - 2)/(x^2 - x - 2). For each function, find the domain and express the answer as an interval or union of intervals.

Learn how to find the domain of the rational functions f(x) = (x - 7)/(x^2 + 14x + 49) and g(x) = (x - 2)/(x^2 - x - 2). Understand how factoring quadratics helps in determining the domain restrictions, suitable for high school students in Grades 10-12.

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