Question

Let $f\left(x\right)=-1+|x-2|,$ and $g\left(x\right)=1-|x|;$ then the set of all points where fog is discontinuous is :

• A. $\left\{0,2\right\}$
• B. $\left\{0,1,2\right\}$
• C. $\left\{0\right\}$
• D. an empty set

Answer 1

Answer: (D)

$fog=f\left(g\left(x\right)\right)=f\left(1-|x|\right)$
$=-1+\left|1-\right|x\left|-2\right|$
$-1+\left|-\right|x\left|-1\right|=-1+\left|x\right|\left|+1\right|$
Let $fog=y$
$\therefore y=-1\left|x\right|\left|+1\right|$
$\Rightarrow y=$ $=\{\begin{array}{ll}-1+x+1,& \text{ x≥0 }\\ -1-x+1,& \text{ x<0 }\end{array}$
$\Rightarrow y=\{\begin{array}{ll}x,& \text{x≥0 }\\ -x,& \text{ x<0}\end{array}$
$LHL$ at $(x=0)=$ $\underset{x\to 0}{\lim }(-x)=0$
$RHL$ at $(x=0)=$ $\underset{x\to 0}{\lim }(-x)=0$
When $x=0$, then $y=0$
Hence, $LHL$ at $(x=0)=RHL$ at $(x=0)=$ value of $y$ at $(x=0)$
Hence $y$ is continuous at $x=0.$
Clearly at all other point y continuous. Therefore, the set of all points where $fog$ is discontinuous is an empty set.