The domain of definition of f(X) = `sin^(-1)(-x^(2))` is

Options

A. `[-1,1]`
B. `[0,1]`
C. `[-1,0]`
D. `[-2,23]`

Explanation

To find the domain of the function \( f(x) = \sin^{-1}(-x^2) \), we need to determine the values of \( x \) for which the expression inside the inverse sine function is valid. The inverse sine function, \( \sin^{-1}(y) \), is defined for \( y \) in the interval \([-1, 1]\).

### Step-by-step Solution:

1. **Set up the inequality**: Since \( f(x) = \sin^{-1}(-x^2) \), we need to ensure that the argument \(-x^2\) lies within the interval \([-1, 1]\). Therefore, we set up the inequality:
\[ -1 \leq -x^2 \leq 1 \]

2. **Solve the left side of the inequality**: Start with the left part of the inequality:
\[ -1 \leq -x^2 \]
Multiplying through by -1 (remember to reverse the inequality):
\[ 1 \geq x^2 \quad \text{or} \quad x^2 \leq 1 \]

3. **Solve the right side of the inequality**: Now consider the right part of the inequality:
\[ -x^2 \leq 1 \]
Again, multiplying through by -1 (reversing the inequality):
\[ x^2 \geq -1 \]
Since \( x^2 \) is always non-negative, this inequality is always true.

4. **Combine the results**: From the first part, we found that:
\[ x^2 \leq 1 \]
This implies:
\[ -1 \leq x \leq 1 \]

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

### Final Answer:
The domain of definition of \( f(x) = \sin^{-1}(-x^2) \) is \([-1, 1]\).