The value of the function `f(x)=3sinsqrt((pi^(2))/(16)-x^(2))` is defined what `x in `

Options

A. `[-(pi)/(4),(pi)/(4)]`
B. `[0,(3)/(sqrt(2))]`
C. `(-3,3)`
D. none of these

Answer

To determine the values of \( x \) for which the function \( f(x) = 3 \sin \left( \sqrt{\frac{\pi^2}{16} - x^2} \right) \) is defined, we need to ensure that the expression inside the square root is non-negative.

### Step-by-Step Solution:

1. **Identify the condition for the square root:**
The function \( f(x) \) is defined when the expression inside the square root is greater than or equal to zero:
\[ \frac{\pi^2}{16} - x^2 \geq 0 \]

2. **Rearranging the inequality:**
We can rearrange the inequality to isolate \( x^2 \):
\[ \frac{\pi^2}{16} \geq x^2 \]

3. **Taking the square root:**
To solve for \( x \), we take the square root of both sides. Remember that taking the square root introduces both positive and negative solutions:
\[ -\sqrt{\frac{\pi^2}{16}} \leq x \leq \sqrt{\frac{\pi^2}{16}} \]

4. **Simplifying the square root:**
The square root of \( \frac{\pi^2}{16} \) simplifies to:
\[ -\frac{\pi}{4} \leq x \leq \frac{\pi}{4} \]

5. **Conclusion:**
Thus, the function \( f(x) \) is defined for:
\[ x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \]

### Final Answer:
The value of the function \( f(x) \) is defined for \( x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \).