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Let f(x)=x^2 and g(x)=sinx , for all x belong to R . Then the set of all x satisfying: (fogogof)(x)=(gogof)(x) , where (fog)(x)=fg(x) , is

Answers (1)

D Deependra Verma

Solution:

$f[g[g[f(x)]]]=g[g[f(x)]]$ $.........(1)$

$LHS=f[g[g(x^2)]]=f[g[sin x^2]]\ \Rightarrow LHS=f[sin(sin x^2)]=sin^2(sin x^2).........(2)$

$RHS=g[g(x^2)]=g[sin (x^2)]=sin(sin x^2)hspace0.5cm.........(3)$

From $(2)$ and $(3)$ , we have

$sin (sin x^2)[sin (sin x^2)-1]=0\ \Rightarrow hspace1cmsin(sin x^2)=0\ \Rightarrow hspace1cmsin x^2=0Rightarrow x^2=n pi,\ \Rightarrow hspace1cmx=pmsqrtnpi, nin �,1,2......$

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