If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=|x|$ and $g(x)=[x-3]$ for $x \in R$,…
If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=|x|$ and $g(x)=[x-3]$ for $x \in R$, then $\left\{g(f(x)):-\frac{8}{5} < x < \frac{8}{5}\right\}$ is equal to
$\{0,1\}$
$\{1,2\}$
$\{-3,-2\}$
$\{2,3\}$
Solution
Given that, $f(x)=|x|$ and $g(x)=[x-3]$
For $\quad-\frac{8}{5} < x < \frac{8}{5}, 0 \leq f(x) < \frac{8}{5}$
Now, for $\quad 0 \leq f(x) < 1$,
$
\begin{aligned}
g(f(x)) & =[f(x)-3] \\
& =-3[\because-3 \leq f(x)-3 < -2]
\end{aligned}
$
Again, for $1 \leq f(x) < 1.6$
$
g(f(x))=-2
$
Hence, required set is $\{-3,-2\}$