The domain of `sin^(-1) 2x` is

Options

A. [0,1]
B. `[-1,1]`
C. `[-1//2,1//2]`
D. `[-2, 2]`

Answer

To find the domain of the function \( \sin^{-1}(2x) \), we need to ensure that the argument of the sine inverse function, which is \( 2x \), lies within the range of the sine inverse function. The sine inverse function, \( \sin^{-1}(y) \), is defined for \( y \) values in the interval \([-1, 1]\).

### Step-by-Step Solution:

1. **Identify the condition for the sine inverse function:**
\[ -1 \leq 2x \leq 1 \]
This means that the value of \( 2x \) must be between -1 and 1.

2. **Split the inequality into two parts:**
- From the left side:
\[ 2x \geq -1 \]
- From the right side:
\[ 2x \leq 1 \]

3. **Solve the left inequality:**
\[ 2x \geq -1 \implies x \geq -\frac{1}{2} \]

4. **Solve the right inequality:**
\[ 2x \leq 1 \implies x \leq \frac{1}{2} \]

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

6. **Write the domain in interval notation:**
The domain of \( \sin^{-1}(2x) \) is:
\[ \boxed{[-\frac{1}{2}, \frac{1}{2}]} \]