If \( y = \sin^{-1}x \), where \( -1 \leq x \leq 0 \), then the range of \( y \) is:
If \( y = \sin^{-1}x \), where \( -1 \leq x \leq 0 \), then the range of \( y \) is:
\( \left( -\frac{\pi}{2}, 0 \right) \)
\( \left[ -\frac{\pi}{2}, 0 \right) \)
\( \left[ -\frac{\pi}{2}, 0 \right] \)
\( \left( -\frac{\pi}{2}, 0 \right] \)
The Correct Option is C
Solution and Explanation
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: \( \boxed{ \left[ -\frac{\pi}{2}, 0 \right] } \)
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