The domain of the function `f (x) = sin^(-1) { log _(2) ((1)/(2) x ^(2)) } is `

Options

A. `[-2, 1) uu [1,2]`
B. `(-2, 1] uu[1,2]`
C. `[-2,-1]uu [1,2]`
D. `(-2,-1)uu (1,2)`

Answer

To find the domain of the function \( f(x) = \sin^{-1} \left( \log_2 \left( \frac{1}{2} x^2 \right) \right) \), we need to ensure that the argument of the inverse sine function is valid. The domain of the function can be determined by following these steps:

### Step 1: Determine the range of the argument for \( \sin^{-1} \)
The function \( \sin^{-1}(y) \) is defined for \( y \) in the interval \([-1, 1]\). Therefore, we need:
\[ -1 \leq \log_2 \left( \frac{1}{2} x^2 \right) \leq 1 \]

### Step 2: Solve the left inequality
Starting with the left inequality:
\[ \log_2 \left( \frac{1}{2} x^2 \right) \geq -1 \]
This can be rewritten using the properties of logarithms:
\[ \frac{1}{2} x^2 \geq 2^{-1} \]
\[ \frac{1}{2} x^2 \geq \frac{1}{2} \]
Multiplying both sides by 2:
\[ x^2 \geq 1 \]
This implies:
\[ x^2 - 1 \geq 0 \]
Factoring gives:
\[ (x - 1)(x + 1) \geq 0 \]
The critical points are \( x = -1 \) and \( x = 1 \). Analyzing the sign changes, we find:
- The solution is valid for \( x \leq -1 \) or \( x \geq 1 \).

### Step 3: Solve the right inequality
Now, we solve the right inequality:
\[ \log_2 \left( \frac{1}{2} x^2 \right) \leq 1 \]
Rewriting gives:
\[ \frac{1}{2} x^2 \leq 2^1 \]
\[ \frac{1}{2} x^2 \leq 2 \]
Multiplying both sides by 2:
\[ x^2 \leq 4 \]
This implies:
\[ x^2 - 4 \leq 0 \]
Factoring gives:
\[ (x - 2)(x + 2) \leq 0 \]
The critical points are \( x = -2 \) and \( x = 2 \). Analyzing the sign changes, we find:
- The solution is valid for \( -2 \leq x \leq 2 \).

### Step 4: Combine the results
Now we combine the results from both inequalities:
1. From the left inequality: \( x \leq -1 \) or \( x \geq 1 \)
2. From the right inequality: \( -2 \leq x \leq 2 \)

The valid intervals that satisfy both conditions are:
- For \( x \leq -1 \): The interval is \( [-2, -1] \).
- For \( x \geq 1 \): The interval is \( [1, 2] \).

### Final Domain
Thus, the domain of the function \( f(x) \) is:
\[ [-2, -1] \cup [1, 2] \]