Question

$f$ : $R\to R,g:R\to R$ are defined as $f\left(x\right)=|x|,g\left(x\right)=\left[x\right]\forall x\in R$ where $\left[x\right]$ denotes greatest integer function. The set of values of $x$ for which $g\left[f\right(x\left)\right]\le f\left[g\right(x\left)\right]$is satisfied is given by

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

Answer 1

Answer: (D)

$R$

For $x\ge 0,f\left(x\right)=x$ and $g\left(x\right)\ge 0$
$g\left(f\right(x\left)\right)=\left[x\right]$ , $f\left(g\right(x\left)\right)=\left[x\right]$ .
Hence $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)$
For $x<0$ , $f\left(x\right)=-x,g\left(x\right)\le -1$
$g\left(f\right(x\left)\right)=[-x],f\left(g\right(x\left)\right)=-\left[x\right]$
Since $x$ is negative, $g\left(f\right(x\left)\right)\le f\left(g\right(x\left)\right)$ (Equality will hold true for integral values, while for non-integral values the difference would be 1)
So the inequality holds true for all values of $x$. Hence D is the correct option