We need to find the domain of the expression:

\[
f(x) = \frac{\sqrt{x-4}}{x^2 - 25}
\]

### Step 1: Constraints from the square root
The square root function \(\sqrt{x-4}\) requires that the expression inside the square root must be non-negative, meaning:
\[
x - 4 \geq 0 \implies x \geq 4
\]
Thus, \(x \geq 4\).

### Step 2: Constraints from the denominator
The denominator \(x^2 - 25\) must not be equal to zero, because division by zero is undefined. So, we solve the equation:
\[
x^2 - 25 = 0 \implies (x - 5)(x + 5) = 0
\]
This gives \(x = 5\) and \(x = -5\). However, \(x = -5\) is irrelevant since \(x \geq 4\), and \(-5\) is not within this range.

Therefore, the only value we exclude due to the denominator is \(x = 5\).

### Step 3: Conclusion
The domain is \(x \geq 4\), excluding \(x = 5\). Thus, the domain of the function is:
\[
\boxed{[4, 5) \cup (5, \infty)}
\]

Would you like more details or have any other questions?

Here are some related questions:
1. How do you find the domain of a rational function?
2. What happens when the denominator of a function is zero?
3. How do square roots affect the domain of a function?
4. What is the difference between the range and domain of a function?
5. How would the domain change if the square root was raised to an even power?

**Tip:** Always check both numerator and denominator constraints separately when determining the domain of a rational function.

Discover how to find the domain of the function sqrt(x-4)/(x^2-25) with a detailed, step-by-step solution. Learn how to apply constraints from both the square root and denominator, ensuring the function is well-defined. Perfect for high school students learning about rational functions and domain restrictions.

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