Question

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

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

Answer 1

The correct option is A all non-integers except the interval $[-1,1]$

$f\left(x\right)=\frac{{\text{cosec}}^{-1}x}{\sqrt{x-\left[x\right]}}$
$\Rightarrow x\in (-\infty ,-1]\cup [1,\infty )$
and $x-\left[x\right]=\left\{x\right\}\ne 0$
$\Rightarrow x\ne$ Integer
$\Rightarrow x\in (-\infty ,-1)\cup (1,\infty )-$all integers