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