To find the domain of the function $f(x) = \frac{-5}{\sqrt{-9 + x}}$, we must ensure that the expression inside the square root, $-9 + x$, is non-negative, because the square root is only defined for non-negative values in the set of real numbers.

Step 1: Set the inequality

To ensure the expression inside the square root is non-negative, we set: $-9 + x \geq 0$

Step 2: Solve the inequality

$x \geq 9$

Step 3: Conclusion for the domain

Since the function involves a square root in the denominator, the value inside the square root cannot be zero either, as that would make the denominator undefined. Thus, we also need to exclude the case where $x = 9$.

Therefore, the domain is $x > 9$. In interval notation, this is: $(9, \infty)$

Would you like any further details or clarifications?

Related Questions:

1. How does the domain change if we have a different constant under the square root?
2. What happens to the domain if the square root is not in the denominator?
3. How does adding or subtracting other terms outside the square root affect the domain?
4. Can you explain how to find the domain of functions involving logarithms?
5. What would be the domain if the function had no square root?

Tip:

Always remember that when a square root is in the denominator, it must be strictly greater than zero to avoid division by zero.