Question

If $f:R\to R$ and $g:R\to R$ are defined by $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=[x-3]$ for $x\in R$, then
$g\left(f\left(x\right)\right):\{-\frac{8}{5}<x<\frac{8}{5}\}$ is equal to

[.] is Greatest integer function

• A. $\{0,1\}$
• B. $\{1,2\}$
• C. $\{-3,-2\}$
• D. $\{2,3\}$

Answer 1

The correct option is C $\{-3,-2\}$

Given that, $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=[x-3]$
For $-\frac{8}{5}<x<\frac{8}{5},0\le f\left(x\right)<\frac{8}{5}$
Now, for $0<f\left(x\right)<1$,
$g\left(f\left(x\right)\right)=[f\left(x\right)-3]$
$=-3\because -3\le f\left(x\right)-3<-2$
Again, for $1<f\left(x\right)<1.6$
$g\left(f\left(x\right)\right)=-2$
$\because -2\le f\left(x\right)-3<-1.4$
Hence, required set is $\{-3,-2\}$