The domain of the function
`f(x)=sqrt(sin^(-1)(log_(2)x))` is

Options

A. `(1,2)`
B. `[1,2)`
C. `(1,2]`
D. none

Answer

To find the domain of the function \( f(x) = \sqrt{\sin^{-1}(\log_2 x)} \), we need to ensure that the expression inside the square root is non-negative and that the argument of the inverse sine function is within its valid range.

### Step 1: Determine the range of the inverse sine function
The function \( \sin^{-1}(y) \) is defined for \( y \) in the range \([-1, 1]\). Therefore, we require:
\[ -1 \leq \log_2 x \leq 1 \]

### Step 2: Solve the inequalities
We will solve the two inequalities separately.

**First inequality:**
\[ \log_2 x \geq -1 \]
This can be rewritten in exponential form:
\[ x \geq 2^{-1} = \frac{1}{2} \]

**Second inequality:**
\[ \log_2 x \leq 1 \]
This can also be rewritten in exponential form:
\[ x \leq 2^1 = 2 \]

### Step 3: Combine the inequalities
From the two inequalities, we have:
\[ \frac{1}{2} \leq x \leq 2 \]

### Step 4: Consider the domain of the logarithm
The logarithm function \( \log_2 x \) is defined only for \( x > 0 \). Since \( \frac{1}{2} \) is greater than 0, the domain condition is satisfied.

### Final Domain
Combining all the conditions, the domain of the function \( f(x) \) is:
\[ x \in \left[\frac{1}{2}, 2\right] \]

### Conclusion
Thus, the domain of the function \( f(x) = \sqrt{\sin^{-1}(\log_2 x)} \) is:
\[ \boxed{\left[\frac{1}{2}, 2\right]} \]