Write the domain of the following real functions
`sin^(-1)[log_(2)((x)/(2))]`

Answer

To find the domain of the function \( f(x) = \sin^{-1} \left[ \log_{2} \left( \frac{x}{2} \right) \right] \), we need to ensure that the argument of the sine inverse function lies within its valid range.

### Step 1: Determine the range of the sine inverse function
The sine inverse function, \( \sin^{-1}(y) \), is defined for \( y \) in the interval \([-1, 1]\). Therefore, we need to set up the inequality:
\[ -1 \leq \log_{2} \left( \frac{x}{2} \right) \leq 1 \]

### Step 2: Solve the left inequality
First, we solve the left part of the inequality:
\[ \log_{2} \left( \frac{x}{2} \right) \geq -1 \]
This can be rewritten in exponential form:
\[ \frac{x}{2} \geq 2^{-1} \]
\[ \frac{x}{2} \geq \frac{1}{2} \]
Multiplying both sides by 2 gives:
\[ x \geq 1 \]

### Step 3: Solve the right inequality
Next, we solve the right part of the inequality:
\[ \log_{2} \left( \frac{x}{2} \right) \leq 1 \]
Rewriting this in exponential form gives:
\[ \frac{x}{2} \leq 2^{1} \]
\[ \frac{x}{2} \leq 2 \]
Multiplying both sides by 2 yields:
\[ x \leq 4 \]

### Step 4: Combine the results
From the two inequalities we derived, we have:
\[ 1 \leq x \leq 4 \]
Thus, the domain of the function \( f(x) \) is:
\[ \text{Domain} = [1, 4] \]

### Final Answer
The domain of the function \( f(x) = \sin^{-1} \left[ \log_{2} \left( \frac{x}{2} \right) \right] \) is:
\[ [1, 4] \]