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We are tasked with finding the domain of the function:

\[
y = \frac{x + 3}{4 - \sqrt{x^2 - 9}}.
\]

To determine the domain of this function, we must ensure that both the denominator is non-zero and the square root is defined. Here's how to proceed:

### Step 1: Analyze the square root
The square root term, \(\sqrt{x^2 - 9}\), must be defined. This means that the expression inside the square root must be non-negative:

\[
x^2 - 9 \geq 0.
\]

Solving this inequality:

\[
x^2 \geq 9 \quad \Rightarrow \quad x \leq -3 \quad \text{or} \quad x \geq 3.
\]

Thus, \(x \in (-\infty, -3] \cup [3, \infty)\).

### Step 2: Ensure the denominator is non-zero
The denominator \(4 - \sqrt{x^2 - 9}\) must not be equal to zero:

\[
4 - \sqrt{x^2 - 9} \neq 0.
\]

This implies:

\[
\sqrt{x^2 - 9} \neq 4.
\]

Squaring both sides:

\[
x^2 - 9 \neq 16 \quad \Rightarrow \quad x^2 \neq 25 \quad \Rightarrow \quad x \neq 5 \quad \text{and} \quad x \neq -5.
\]

### Step 3: Combine the conditions
From Step 1, the domain is \(x \in (-\infty, -3] \cup [3, \infty)\). From Step 2, we exclude \(x = 5\) and \(x = -5\).

Thus, the domain is:

\[
(-\infty, -5) \cup (-5, -3] \cup [3, 5) \cup (5, \infty).
\]

### Final Answer:
The domain of the function is:

\[
x \in (-\infty, -5) \cup (-5, -3] \cup [3, 5) \cup (5, \infty).
\]

Would you like further details or have any questions about this solution?

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Here are 5 related questions to explore:

1. How would the domain change if the denominator was \(4 + \sqrt{x^2 - 9}\)?
2. Can a complex number be part of the domain if the square root is negative?
3. How does squaring both sides in Step 2 affect the solution?
4. What happens to the domain if we remove the square root term entirely?
5. How would the solution change if the expression in the denominator was \(4 - \sqrt{9 - x^2}\)?

**Tip:** Always check for values that make the denominator zero or the square root term negative when finding the domain of a rational function involving square roots.

Find the domain of y = (x + 3) / (4 - sqrt(x^2 - 9)).

This guide explores how to find the domain of the rational function y = (x + 3) / (4 - sqrt(x^2 - 9)). The solution involves analyzing square root expressions and avoiding division by zero. Suitable for high school students in grades 10-12.

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