The domain cannot include \( x = -4 \) because it would make the denominator of the function equal to zero, which results in an undefined expression.

Here's why:

For the function \( f(x) = \frac{x^2 - 1}{x + 4} \):

1. **The denominator is \( x + 4 \)**. If \( x = -4 \), the denominator becomes:
   \[
   x + 4 = -4 + 4 = 0
   \]
2. **Division by zero is undefined** in mathematics. When the denominator is zero, the function does not produce a valid number because there is no number that can be the result of division by zero. Hence, the function is undefined at \( x = -4 \).

This is why \( x = -4 \) must be excluded from the domain of the function.

Would you like more details or have any further questions?

Here are 5 related questions:

1. What happens if the numerator of a rational function is zero?
2. How do vertical asymptotes relate to the domain of rational functions?
3. How do you handle division by zero in piecewise functions?
4. How do you find the domain when the denominator has a higher-degree polynomial?
5. What are the consequences of having both the numerator and denominator equal to zero?

**Tip:** Always set the denominator of a rational function not equal to zero when determining the domain.

Understand why x = -4 must be excluded from the domain of the function f(x) = (x^2 - 1) / (x + 4). Explore the concept of division by zero and learn how to determine the domain of rational functions. Suitable for high school students in Grades 9-12.

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