The function you've provided is:

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

This expression has a square root in the denominator, which implies certain restrictions on the domain of \( f(x) \). Let's analyze the function in more detail.

### Domain Restrictions:
1. The expression inside the square root, \( 9 - 4x^2 \), must be non-negative (i.e., \( 9 - 4x^2 \geq 0 \)) for the square root to be real.
2. \( 9 - 4x^2 \geq 0 \) simplifies to \( 9 \geq 4x^2 \), or \( \frac{9}{4} \geq x^2 \). Taking the square root on both sides gives \( \frac{3}{2} \geq |x| \), which simplifies to \( -\frac{3}{2} \leq x \leq \frac{3}{2} \).

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

Would you like to further analyze this function, such as finding its derivative or identifying asymptotes?

### Follow-up Questions:
1. Do you want to simplify or rewrite the function in another form?
2. Would you like to find the critical points of \( f(x) \) by taking its derivative?
3. Are you interested in finding the local minima or maxima of this function?
4. Do you want to investigate the behavior of \( f(x) \) as \( x \) approaches the boundaries of its domain?
5. Would you like to graph the function for a visual representation?

#### Tip:
Always check the domain of functions with square roots to ensure you're working within the real number system.

Analyze the function f(x) = 6x / √(9 - 4x^2), focusing on the domain, critical points, and behavior at the domain boundaries. Learn how to determine the domain of radical functions and explore their properties in Grades 10-12.

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