Question

The real valued function f(x) = csc⁻¹x / √(x - [x]), where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :

• A. all reals except integers
• B. all reals except the interval [-1, 1]
• C. all non-integers except the interval (-1, 1)
• D. all integers except 0, -1, 1

Hint:

The fractional part function $\{x\} = x - [x]$ always satisfies $0 \le \{x\}<1$. It is zero at every integer.

Answer 1

The Correct Option is C

Step 1: For $\csc^{-1}x$, the domain is $|x| \ge 1$. This means $x \in (-\infty, -1] \cup [1, \infty)$.
Step 2: For the denominator $\sqrt{x - [x]}$, the term inside the square root must be positive. Let $\{x\} = x - [x]$ be the fractional part. We need $\{x\}>0$.
Step 3: The fractional part $\{x\}$ is zero if and only if $x$ is an integer ($x \in \mathbb{Z}$). Thus, $x \notin \mathbb{Z}$.
Step 4: Combining these: $x \in ((-\infty, -1] \cup [1, \infty))$ AND $x \notin \mathbb{Z}$. This simplifies to all non-integers except the interval $(-1, 1)$.