Explanation
Solution
To find the domain, we must satisfy the conditions for both the inner sin−1 function and the outer cos−1 function.
1. Condition for the inner function sin−1(y)
The argument of sin−1 must lie between −1 and 1:
Also, the denominator cannot be zero: 4x2−1=0⟹x=±21.
Case A: 4x2−11≤1
4x2−11−(4x2−1)≤0⟹4x2−12−4x2≤0
Using the critical points ±21 and ±21, we find:
x∈(−∞,−21]∪(−21,21)∪[21,∞)
Case B: 4x2−11≥−1
4x2−11+4x2−1≥0⟹4x2−14x2≥0
Since 4x2 is always ≥0, we just need 4x^2 - 1 > 0 or x=0.
Intersection of Case A and B:
x∈(−∞,−21]∪[21,∞)∪{0}
2. Condition for the outer function cos−1(z)
The argument of cos−1 must also lie between −1 and 1:
Multiplying by 2π:
Since the range of sin−1 is already [−2π,2π], this condition is always satisfied for any x that is valid for the inner sin−1 function.
Final Conclusion
Combining the results, the domain is:
x∈(−∞,−21]∪[21,∞)∪{0}
Correct Option: (D)
