Question

If \( y = \sin^{-1}x \), where \( -1 \leq x \leq 0 \), then the range of \( y \) is: 

• A. \( \left( -\frac{\pi}{2}, 0 \right) \)
• B. \( \left[ -\frac{\pi}{2}, 0 \right) \)
• C. \( \left[ -\frac{\pi}{2}, 0 \right] \)
• D. \( \left( -\frac{\pi}{2}, 0 \right] \)

Answer 1

The Correct Option is C

Step 1: Understanding the Function

The function \( y = \sin^{-1}x \) is the inverse sine function. 

Its domain is: \( [-1, 1] \)
Its range is: \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)

Step 2: Restrict the Domain

In the given problem, the domain is restricted to: \( -1 \leq x \leq 0 \)

Step 3: Finding the Range Corresponding to the Given Domain

• At \( x = -1 \), \( y = \sin^{-1}(-1) = -\frac{\pi}{2} \)
• At \( x = 0 \), \( y = \sin^{-1}(0) = 0 \)

Since the inverse sine function is continuous and strictly increasing, the range of \( y \) is:

\[ \left[ -\frac{\pi}{2}, 0 \right] \]

Step 4: Conclusion

Correct Answer: \( \boxed{ \left[ -\frac{\pi}{2}, 0 \right] } \)