Limits Continuity And Differentiability Question 119
Question: Let $ f’(x)=-1+| x-2 |, $ and $ g(x)=1-| x |; $ then the set of all points where fog is discontinuous is:
Options:
A) {0, 2}
B) {0, 1, 2}
C) {0}
D) An empty set
Show Answer
Answer:
Correct Answer: D
Solution:
$ f(g)(x))=f(1-| x |)=-1+| | x |+1 | $ Let $ fog=y $
$ \therefore y=-1+| | x |+1 |\Rightarrow y= \begin{matrix} x, & x\ge 0 \\ -x, & x<0 \\ \end{matrix} . $ LHL at $ (x=0)=\underset{x\to 0}{\mathop{\lim }}(-x)=0 $ RHL at $ (x=0)=\underset{x\to 0}{\mathop{\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.
BETA
