What is the domain of the function `y= "sin"^(-1)(-x^2).`

Options

A. `-1 le x le 1`
B. `- infty le x le infty`
C. ` x le 1`
D. `x ge 0`

Answer

To find the domain of the function \( y = \sin^{-1}(-x^2) \), we need to determine the values of \( x \) for which the function is defined. The inverse sine function, \( \sin^{-1}(x) \), has a specific range for its argument, which must be between -1 and 1 (inclusive).

### Step-by-Step Solution:

1. **Set up the inequality for the inverse sine function:**
\[ -1 \leq -x^2 \leq 1 \]

2. **Rearranging the inequality:**
- First, we can multiply the entire inequality by -1. Remember that when we multiply or divide an inequality by a negative number, we must reverse the inequality signs:
\[ 1 \geq x^2 \geq -1 \]
- Since \( x^2 \) is always non-negative, the condition \( x^2 \geq -1 \) is always satisfied. Therefore, we only need to consider:
\[ x^2 \leq 1 \]

3. **Solving for \( x \):**
- The inequality \( x^2 \leq 1 \) can be rewritten as:
\[ -1 \leq x \leq 1 \]

4. **Conclusion:**
- The domain of the function \( y = \sin^{-1}(-x^2) \) is:
\[ [-1, 1] \]