# The real valued function

Question:

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

1. (1) all reals except integers

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

3. (3) all integers except $0,-1,1$

4. (4) all reals except the Interval $[-1,1]$

Correct Option: , 2

Solution:

$f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{\{x\}}}$

Domain $\in(-\infty,-1] \cup[1, \infty)$

$\{x\} \neq 0$ so $x \neq$ integers