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The given function is:

\[
f(x) = \frac{10}{\sqrt{x^2 - 9}}
\]

### Domain of the Function:
To determine the domain of the function, we must consider the conditions under which the square root and the denominator are defined.

1. The expression under the square root, \(x^2 - 9\), must be greater than 0 because the square root of a negative number is not real and division by 0 is undefined.

   \[
   x^2 - 9 > 0
   \]

   Solving this inequality:

   \[
   x^2 > 9
   \]

   Taking the square root of both sides:

   \[
   |x| > 3 \quad \text{or} \quad x > 3 \quad \text{or} \quad x < -3
   \]

   Therefore, the domain of \(f(x)\) is:

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

Would you like further details or clarifications? Here are 5 questions to expand on this:

1. What happens to the function as \(x\) approaches 3 from the right or left?
2. How would the graph of this function look?
3. How does the function behave for large values of \(x\)?
4. What is the range of the function?
5. How would you find the derivative of this function?

**Tip:** Always remember that when dealing with square roots in a function's denominator, ensure the expression inside is positive to avoid undefined values.

Learn how to find the domain of the function f(x) = 10 / sqrt(x^2 - 9). This solution covers key concepts such as inequalities and square root properties. Suitable for students in Grades 10-12.

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