Question

Let $[x]$ denote the greatest integer not more than $x$. If $A$ and $B$ are the domains of the functions $f(x)=\frac{x-[x]}{\sqrt{|x|-x}}$ and $g(x)=\frac{x-[x]}{\sqrt{|x|+x}}$ respectively, then

• A. $A\cup B=R$
• B. $A\cap B=\varphi$
• C. $A-B=(-\infty ,0)$
• D. $B-A=(0,\infty )$

Answer 1

Answer: (B)

Given function, $f(x)=\frac{x-[x]}{\sqrt{|x|-x}}$ will define, if $|x|>x\Rightarrow x<0\Rightarrow x\in (-\infty ,0)\dots$ (i)
and the function, $g(x)=\frac{x-[x]}{\sqrt{|x|+x}}$ will define, if
$|x|+x>0\Rightarrow x>0\Rightarrow x\in (0,\infty )\dots$ (ii)
$\therefore A=\{x\mid x\in R$ and $x<0\}$
and $B=\{x\mid x\in R$ and $x>0\}$
$\therefore A\cap B=\varphi$