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Question

Let f(x)=x2 and g(x) =sin x for all xR. Then the set of all x satisfying (fogogof) (x) =(gogof) (x), where (fog) (x) =f(g(x)), is 
 
  1. 2nπ,n{....,2,1,0,1,2,....}
     
  2. ±nπ,n{1,2,....}
     
  3. ±nπ,n{0,1,2,....}
     
  4. π2+2nπ,nin{....,2,1,0,1,2,....}
     


Solution

The correct option is C ±nπ,n{0,1,2,....}
 

(fogogof)(x)=sin2(sinx2)(gogof)(x)=sin(sinx2)sin2(sinx2)=sin(sinx2)sin(sinx2)[sin(sinx2)1]=0sin(sinx2)=0or1
sinx2=nπor2mπ+π2, where m,nI
sinx2=0x2=nπx=±nπ,n{0,1,2,....}

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