To find the domain of the function \( f(x) = \frac{3x}{\sqrt{x + 5}} \), we need to determine the values of \( x \) for which the function is defined. There are two conditions to consider:

1. The denominator cannot be zero. So, we need to ensure that \( \sqrt{x + 5} \neq 0 \).
2. The expression inside the square root must be non-negative, since the square root of a negative number is undefined for real numbers. So, \( x + 5 \geq 0 \).

### Step-by-step:

1. **Denominator condition**:  
   \[
   \sqrt{x + 5} \neq 0 \quad \Rightarrow \quad x + 5 \neq 0 \quad \Rightarrow \quad x \neq -5
   \]

2. **Non-negative square root condition**:  
   \[
   x + 5 \geq 0 \quad \Rightarrow \quad x \geq -5
   \]

### Conclusion:
The domain of the function \( f(x) = \frac{3x}{\sqrt{x + 5}} \) is all real numbers \( x \) such that \( x > -5 \). In interval notation, the domain is:
\[
(-5, \infty)
\]

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

### Related Questions:
1. What happens if we have a different function, like \( f(x) = \frac{3}{\sqrt{x + 2}} \)?
2. How would we handle domain restrictions for rational functions?
3. What is the domain of a function involving cube roots?
4. How can we find the range of functions with square roots?
5. Can a function have more than one domain restriction?

### Tip:
When dealing with square roots in the denominator, always check both for non-negativity (inside the root) and that the denominator does not become zero.

Learn how to find the domain of the function f(x) = 3x / sqrt(x + 5). This step-by-step solution covers domain restrictions for square roots, including non-negativity and avoiding zero in the denominator. Suitable for students in Grades 9-11.

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