The function given is:

\[
f(x) = \frac{6x}{\sqrt{9 - 4x^2}}
\]

This expression is already in its simplest form. However, if you'd like to see more details, such as the domain or its derivative, here's some further insight:

### 1. **Domain of the function**:
For the function \( f(x) \) to be defined, the expression inside the square root must be non-negative because the square root of a negative number is not real. So, we need:

\[
9 - 4x^2 \geq 0
\]

Solving this inequality:

\[
9 \geq 4x^2
\]
\[
\frac{9}{4} \geq x^2
\]
\[
x^2 \leq \frac{9}{4}
\]
\[
|x| \leq \frac{3}{2}
\]

Therefore, \( x \in \left[-\frac{3}{2}, \frac{3}{2}\right] \).

Thus, the domain of \( f(x) \) is \( \left[-\frac{3}{2}, \frac{3}{2}\right] \).

### 2. **Derivative of the function**:
If you'd like the derivative, let me know, and I can compute it for you!

Would you like more details or have any questions?

Here are some related questions for further practice:
1. What is the derivative of \( f(x) \)?
2. How would you find the critical points of this function?
3. How can you graph \( f(x) \)?
4. What is the behavior of \( f(x) \) at the boundary points \( x = \pm \frac{3}{2} \)?
5. Can you express \( f(x) \) as a composition of two functions for chain rule purposes?

**Tip**: Always check the domain of a function when it involves square roots or divisions to ensure it is well-defined.

Explore how to find the domain of the function f(x) = 6x / √(9 - 4x^2), covering key concepts such as solving inequalities and understanding square root functions. This is ideal for students in Grades 10-12.

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