To find the domain of the function f(x), we must satisfy the conditions for both the numerator and the denominator simultaneously.
1. Condition for the Numerator (cos−1x):
The inverse cosine function, cos−1x, is defined only when its input x lies in the interval:
So, from the numerator, we have x∈[−1,1].
2. Condition for the Denominator ([x]):
The function f(x) is defined only when the denominator is non-zero. Here, [x] represents the Greatest Integer Function.
We know that [x]=0 for all x in the interval [0,1). Therefore, to satisfy the condition [x]=0, we must exclude the interval [0,1) from our domain.
3. Combining both conditions:
We need the intersection of the results from Step 1 and Step 2:
Intersection:
Since the only value in the second part of the union is exactly 1, we can write it as:
Conclusion:
The domain of the function is the set of all x such that -1 \leq x < 0 or x=1.
Correct Option:
A) [−1,0)∪{1}