1)

Let f(x)= x2 and g(x)= sinx for all x ϵ R. Then, the set of all x satisfying (fogogof)(x) = (gogof)(x), where (fog)(x) = f(g(x))is


A) ±nπ,nϵ{0,1,2....}

B) ±nπ,nϵ{1,2....}

C) π2+2nπ,nϵ{.....2,1,0,1,2....}

D) 2nπ,nϵ{.....2,1,0,1,2....}

Answer:

Option B

Explanation:

 f(x)=x2,g(x)=sinx

 (gof)(x)=sinx2

 go(gof)(x)=sin(sinx2)

 (fogogof)(x)=(sin(sinx2))2.......(i)

 Again  , (gof)(x)=sinx2

 (gogof)(x)=sin(sinx2).....(ii)

 Given,  (fogogof)(x)=(gogof)(x)

  (sin(sinx2))2=sin(sinx2)

   sin(sinx2)(sin(sinx2)1)=0

    sin(sinx2)=0 or  sin(sinx2)=1

   sinx2=0   or sinx2=π2

   x2=nπ

  (i.e, not possible as 1sinθ1)

 x=±nπ