JEE Main & Advanced JEE Main Paper (Held On 22 April 2013)

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

    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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