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JEE Main2013MathematicsContinuity and DifferentiabilityActual

Let f(x)=-1+|x-2| , and g(x)=1-|x| ; then the set of all points where f_ o g is discontinuous is :

Options

  1. A0,2
  2. B0,1,2
  3. C0
  4. Dan empty set

Correct answer

D. an empty set

Step-by-step solution

aligned & f g=f(g(x))=f(1-|x|) & =-1+|1-| x|-2| & =-1+|-| x|-1|=-1+|| x|+1| aligned Let f g=y aligned & y=-1+|| x|+1| & y= cases -1+x+1, & x 0 -1-x+1, & x < 0 cases & y= array cc x, & x 0 -x, & x < 0 array . aligned LHL at (x=0)= _ x 0 (-x)=0 RHL at (x=0)= _ x 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.

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