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To find the domain of the function \( f(x) = \frac{9}{x + 9} + \frac{3}{x - 5} \), we need to determine when the function is undefined, specifically where the denominators become zero, as division by zero is undefined.

### Steps:
1. For the first term \( \frac{9}{x + 9} \), the denominator is \( x + 9 \). To avoid division by zero:
   \[
   x + 9 \neq 0 \quad \Rightarrow \quad x \neq -9
   \]
2. For the second term \( \frac{3}{x - 5} \), the denominator is \( x - 5 \). To avoid division by zero:
   \[
   x - 5 \neq 0 \quad \Rightarrow \quad x \neq 5
   \]

### Domain:
Thus, the function is undefined at \( x = -9 \) and \( x = 5 \). Therefore, the domain of the function is all real numbers except \( x = -9 \) and \( x = 5 \).

### Final Answer:
\[
\boxed{x \in (-\infty, -9) \cup (-9, 5) \cup (5, \infty)}
\]

Let me know if you'd like more details or have any other questions!

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Here are 5 related questions:
1. How would the domain change if the function was \( f(x) = \frac{9}{(x + 9)(x - 5)} \)?
2. What is the range of the given function?
3. How do you find the vertical asymptotes of rational functions?
4. What are the steps to find horizontal asymptotes for rational functions?
5. How do you solve for the x-intercepts of a rational function?

**Tip:** When working with rational functions, always check for values that make the denominator zero to determine the domain.

Find the domain of the function f(x) = 9/(x + 9) + 3/(x - 5).

Determine the domain of the rational function f(x) = 9/(x + 9) + 3/(x - 5). This step-by-step guide explains how to find values that make the denominator zero and how to define the domain of a function. Suitable for students in Grades 9-11.

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