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Let's find the domain of each function based on the given definitions.

### 1. Domain of \( f(x) = \frac{x^2}{x + 4} \)

The domain of a rational function is all real numbers except where the denominator equals zero, as division by zero is undefined. Here, the denominator is \( x + 4 \).

Set the denominator equal to zero:
\[
x + 4 = 0
\]
\[
x = -4
\]
So, the function \( f(x) \) is undefined at \( x = -4 \). Therefore, the domain of \( f(x) \) is all real numbers except \( -4 \).

**Domain of \( f(x) \):** \( (-\infty, -4) \cup (-4, \infty) \)

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### 2. Domain of \( g(x) = \frac{x + 7}{x^2 - 49} \)

Again, the domain of a rational function is all real numbers except where the denominator is zero. Here, the denominator is \( x^2 - 49 \), which can be factored as:
\[
x^2 - 49 = (x - 7)(x + 7)
\]
Set the denominator equal to zero:
\[
(x - 7)(x + 7) = 0
\]
This gives \( x = 7 \) and \( x = -7 \), meaning the function \( g(x) \) is undefined at these points.

**Domain of \( g(x) \):** \( (-\infty, -7) \cup (-7, 7) \cup (7, \infty) \)

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Would you like further details or have any questions on these domain calculations?

### Related questions:
1. How do you handle functions with more complex denominators?
2. What are the general steps to finding the domain of a composite function?
3. How would you find the range of the functions \( f(x) \) and \( g(x) \)?
4. What happens to the domain when there are square roots involved?
5. Can vertical asymptotes be inferred from the domain restrictions?

### Tip:
When finding the domain of a rational function, always look for values that make the denominator zero, and exclude those from the domain.

The functions f and g are defined as follows: f(x) = x^2 / (x + 4) and g(x) = (x + 7) / (x^2 - 49). For each function, find the domain. Write each answer as an interval or union of intervals.

Learn how to find the domain of the rational functions f(x) = x^2 / (x + 4) and g(x) = (x + 7) / (x^2 - 49). This problem involves factoring and identifying points where the denominator equals zero. Suitable for Grades 9-11.

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