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.