Find the domain of the function : `f(x)=sin^(-1)((log)_2x)`

Explanation

To find the domain of the function \( f(x) = \sin^{-1}(\log_2 x) \), we need to consider the constraints imposed by both the logarithmic function and the inverse sine function.

### Step-by-Step Solution:

1. **Identify the range of the inverse sine function**:
The function \( \sin^{-1}(y) \) is defined for \( y \) in the interval \([-1, 1]\). Therefore, we need:
\[ -1 \leq \log_2 x \leq 1 \]

2. **Convert the logarithmic inequalities to exponential form**:
We can rewrite the inequalities involving the logarithm:
- From \( \log_2 x \geq -1 \):
\[ x \geq 2^{-1} = \frac{1}{2} \]
- From \( \log_2 x \leq 1 \):
\[ x \leq 2^1 = 2 \]

3. **Combine the inequalities**:
Combining the results from the two inequalities, we get:
\[ \frac{1}{2} \leq x \leq 2 \]

4. **Write the domain in interval notation**:
The domain of the function \( f(x) \) is:
\[ \left[\frac{1}{2}, 2\right] \]

### Final Answer:
The domain of the function \( f(x) = \sin^{-1}(\log_2 x) \) is \( \left[\frac{1}{2}, 2\right] \).

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