The real function $f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{x-[x]}}$, where $[x]$ denotes the greatest integer less than or equal to $x$, is defined for all $x$ belonging to :

(B) all non-integers except the interval $[-1,1]$

(A) all integers except $0,-1,1$

(C) all reals except the interval $[-1,1]$

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Sets Relations and Functions

The real function

f (x) = \frac{cosec ^{- 1} x}{x - [ x ]}

, where

[x]

denotes the greatest integer less than or equal to

x

, is defined for all

x

belonging to :

Answer: (B)

Two conditions must hold. The inverse cosecant is defined only for

∣ x ∣ \geq 1

, so

x \in (- \infty, - 1] \cup [1, \infty)

. The denominator

x - [x]

is the square root of the fractional part of

x

, which is never negative and vanishes exactly at the integers, so

x

must not be an integer. Since

x = \pm 1

are themselves integers, the two conditions together leave every non-integer lying outside the interval

[- 1, 1]

, and that set is the domain of

f

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Single correct option