The domain of definition of `f(x)=sin^(- 1)[2-4x^2]` is ([.] denotes the greatest integer function).

Options

A. `[sqrt3//2, sqrt3//2]`
B. `[-sqrt3//2, 0]`
C. `[-sqrt3//2, 0) uu(0, sqrt3//2]`
D. `[-sqrt3//2, oo)`

Explanation

To find the domain of the function \( f(x) = \sin^{-1}[2 - 4x^2] \), we need to ensure that the argument of the inverse sine function lies within the valid range, which is between -1 and 1, inclusive. Additionally, we need to consider the greatest integer function (denoted by square brackets).

### Step-by-Step Solution:

1. **Identify the range for the argument of \( \sin^{-1} \)**:
\[ -1 \leq 2 - 4x^2 \leq 1 \]

2. **Break this into two inequalities**:
- First inequality:
\[ 2 - 4x^2 \geq -1 \]
- Second inequality:
\[ 2 - 4x^2 \leq 1 \]

3. **Solve the first inequality**:
\[ 2 - 4x^2 \geq -1 \implies 3 \geq 4x^2 \implies \frac{3}{4} \geq x^2 \]
This simplifies to:
\[ x^2 \leq \frac{3}{4} \]

4. **Solve the second inequality**:
\[ 2 - 4x^2 \leq 1 \implies 1 \leq 4x^2 \implies \frac{1}{4} \leq x^2 \]
This simplifies to:
\[ x^2 \geq \frac{1}{4} \]

5. **Combine the results**:
From the two inequalities, we have:
\[ \frac{1}{4} \leq x^2 \leq \frac{3}{4} \]

6. **Express in terms of \( x \)**:
Taking the square root gives:
\[ \sqrt{\frac{1}{4}} \leq |x| \leq \sqrt{\frac{3}{4}} \]
This translates to:
\[ \frac{1}{2} \leq |x| \leq \frac{\sqrt{3}}{2} \]

7. **Write the domain**:
This means:
\[ -\frac{\sqrt{3}}{2} \leq x \leq -\frac{1}{2} \quad \text{or} \quad \frac{1}{2} \leq x \leq \frac{\sqrt{3}}{2} \]

8. **Final domain representation**:
Therefore, the domain of \( f(x) \) is:
\[ x \in \left[-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right] \cup \left[\frac{1}{2}, \frac{\sqrt{3}}{2}\right] \]