Inverse Trigonometric Functions

To find the domain of the function f(x)=sin‌−1(log2(‌

x2

2

)), we need to determine where the expression log2(‌

x2

2

) falls within the domain of the inverse sine function, which is [−1,1].
First, ensure that:
−1≤log2(‌

x2

2

)≤1
This inequality translates to:
2−1≤‌

x2

2

≤2
Solving the inequality:
‌

1

2

≤‌

x2

2

≤2
Multiply through by 2 to clear the fraction:
1≤x2≤4
This implies that x must satisfy:
x∈[−2,−1]∪[1,2]
We also need to ensure that ‌

x2

2

≥0, which is always true since x2≥0.
Therefore, considering the intersection of the feasible solutions:
x∈[−2,−1]∪[1,2]
Thus, the domain of the function is:
x∈[−2,−1]∪[1,2]