A) {0, 2}
B) {0, 1, 2}
C) {0}
D) an empty set
Correct Answer: D
Solution :
\[fog=f(g(x))=f(1-|x|)\] \[=-1+|1-|x|-2|\] \[=-1+|-|x|-1|=-1+||x|+1|\] Let \[fog=y\] \[\therefore \] \[y=-1+||x|+1|\,\] \[\Rightarrow \]\[y=\left\{ \begin{matrix} -1+x+1, & x\ge 0 \\ -1-x+1, & x<0 \\ \end{matrix} \right.\] \[\Rightarrow \]\[y=\left\{ \begin{matrix} x, & x\ge 0 \\ -x, & x<0 \\ \end{matrix} \right.\] LHL at \[(x=0)=\underset{x\to 0}{\mathop{\lim }}\,(-x)=0\] RHL at \[(x=0)=RHL\]at \[(x=0)=\]value of y at \[(x=0)\] 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.
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