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Let's analyze the given functions to determine their domains.

1. **Function \( f(x) = \frac{x^2}{x + 1} \)**:
   - The domain of a rational function is all real numbers except where the denominator equals zero. For \( f(x) \), the denominator is \( x + 1 \). Setting it to zero:
     \[
     x + 1 = 0 \implies x = -1
     \]
   - So, the domain of \( f(x) \) is all real numbers except \( x = -1 \).
     \[
     \text{Domain of } f(x): (-\infty, -1) \cup (-1, \infty)
     \]

2. **Function \( g(x) = \frac{x - 8}{x^2 - 64} \)**:
   - Again, the domain is all real numbers except where the denominator equals zero. The denominator \( x^2 - 64 \) is a difference of squares and can be factored:
     \[
     x^2 - 64 = (x - 8)(x + 8)
     \]
   - Set each factor equal to zero:
     \[
     x - 8 = 0 \implies x = 8
     \]
     \[
     x + 8 = 0 \implies x = -8
     \]
   - So, the domain of \( g(x) \) is all real numbers except \( x = 8 \) and \( x = -8 \).
     \[
     \text{Domain of } g(x): (-\infty, -8) \cup (-8, 8) \cup (8, \infty)
     \]

### Summary:
- **Domain of \( f(x) \)**: \( (-\infty, -1) \cup (-1, \infty) \)
- **Domain of \( g(x) \)**: \( (-\infty, -8) \cup (-8, 8) \cup (8, \infty) \)

Would you like further details or clarifications? Here are 5 related questions:

1. How do you handle rational functions with more complex denominators?
2. What happens to the domain if a rational function has both a numerator and denominator that can be simplified?
3. Can we use graphical methods to determine domains?
4. What’s the significance of excluded values in a function's domain?
5. How do vertical asymptotes relate to domain restrictions?

**Tip**: Always factor the denominator of a rational function to find all possible values to exclude from the domain.

For the functions f(x) = x^2 / (x + 1) and g(x) = (x - 8) / (x^2 - 64), find the domain of each function and express it as an interval or union of intervals.

Learn how to find the domain of rational functions like f(x) = x^2 / (x + 1) and g(x) = (x - 8) / (x^2 - 64). This problem explores key algebraic concepts such as factoring and handling domain restrictions. Suitable for students in Grades 10-12.

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