If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$,…
If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$, then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to
$Z \cup(-\infty, 0)$
$(-\infty, 0)$
$Z$
$R$
Solution
We have,
$f(x)=|x| \text { and } g(x)=[x]$
Now, $\quad g(f(x)) \leq f(g(x))$
$\Rightarrow \quad g(|x|) \leq f([x])$
$\therefore \quad[|x|] \leq|[x]| \forall x \in R$