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The problem provided asks for the domain of two functions, \( f(x) = \frac{x}{x^2 + 4} \) and \( g(x) = \frac{x + 9}{x^2 - 81} \), expressed as intervals.

### 1. Domain of \( f(x) \):
The domain of a rational function is defined by the values of \( x \) for which the denominator is not zero. For \( f(x) \):

\[
f(x) = \frac{x}{x^2 + 4}
\]

Here, the denominator \( x^2 + 4 \) is always positive (since \( x^2 + 4 \geq 4 \) for all real \( x \)). Therefore, there are no restrictions on \( x \), meaning the domain of \( f(x) \) is all real numbers.

\[
\text{Domain of } f(x): (-\infty, \infty)
\]

### 2. Domain of \( g(x) \):
For \( g(x) \):

\[
g(x) = \frac{x + 9}{x^2 - 81}
\]

The denominator \( x^2 - 81 = (x - 9)(x + 9) \) equals zero when \( x = 9 \) or \( x = -9 \). These values must be excluded from the domain since division by zero is undefined.

\[
\text{Domain of } g(x): (-\infty, -9) \cup (-9, 9) \cup (9, \infty)
\]

### Final Answer:
- Domain of \( f(x) \): \( (-\infty, \infty) \)
- Domain of \( g(x) \): \( (-\infty, -9) \cup (-9, 9) \cup (9, \infty) \)

Would you like any further clarification?

Here are five follow-up questions to deepen your understanding:
1. Why is the domain of \( f(x) \) unrestricted despite having a rational function?
2. How do you factor the denominator of \( g(x) \) to identify critical points?
3. How would the domain change if the function included a square root?
4. What would happen to the domain of \( g(x) \) if the numerator was also zero at the excluded points?
5. How do you write a union of intervals in interval notation?

**Tip:** Always check the denominator for zero when finding the domain of rational functions.

For each function, find the domain. Write each answer as an interval or union of intervals for f(x) = x / (x^2 + 4) and g(x) = (x + 9) / (x^2 - 81).

Discover how to find the domain of the rational functions f(x) = x / (x^2 + 4) and g(x) = (x + 9) / (x^2 - 81). This problem covers key concepts like factoring and identifying excluded values to determine valid intervals for the domain. Ideal for students in Grades 9-11.

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