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{cases} -1+x+1, & \text{ $x \ge 0$ } \\[2ex] -1-x+1, & \text{ $x < 0$ } \end{cases}$
$\Rightarrow y = \begin{cases} x, & \text{$x \ge 0$ } \\[2ex] -x, & \text{ $x < 0$} \end{cases}$
$LHL$ at $(x = 0) =$ $\displaystyle \lim_{x \to 0} (-x)=0$
$RHL$ at $(x = 0) =$ $\displaystyle \lim_{x \to 0} (-x)=0$
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.