Question

Domain of $\cos^{-1}[x]$ is, where $[.]$ denotes greatest integer function:

• A. $(-1, 2]$
• B. $[-1, 2]$
• C. $(-1, 2)$
• D. $[-1, 2)$

Hint:

The greatest integer function $[x] = n$ implies $n \le x <n+1$. Always remember that the upper bound in the domain of $[x]$ results in an open bracket ($2$ is not included because $[2]=2$, which is outside the range $[-1, 1]$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The domain of the function $\cos^{-1}(u)$ is defined for $-1 \le u \le 1$. In this problem, the argument $u$ is the greatest integer function $[x]$. We must find the range of $x$ values such that $[x]$ stays within the interval $[-1, 1]$.

Step 2: Detailed Explanation:
1. Set the condition for the inverse cosine function: \[ -1 \le [x] \le 1 \] 2. Since $[x]$ must be an integer, the possible values for $[x]$ are $\{-1, 0, 1\}$. 3. Analyze each integer value: - If $[x] = -1$, then $-1 \le x <0$. - If $[x] = 0$, then $0 \le x <1$. - If $[x] = 1$, then $1 \le x <2$. 4. Combine these intervals: \[ [-1, 0) \cup [0, 1) \cup [1, 2) = [-1, 2) \]

Step 3: Final Answer
The domain is the interval $[-1, 2)$.