Question

Let `f(x)=x^2a n dg(x)=sinxfora l lx in Rdot` Then the set of all `x` satisfying `(fogogof)(x)=(gogof)(x),w h e r e(fog)(x)=f(g(x)),` is `+-sqrt(npi),n in {0,1,2, dot}` `+-sqrt(npi),n in {1,2, dot}` `pi/2+2npi,n in { ,-2,-1,0,1,2}` `2npi,n in { ,-2,-1,0,1,2, }`

Answer 1

`f(x)=x^2`
`g(x)=sinx`
`g(f(x))=sin(f(x))`
`g@f=sinx^2`
`g(g@f)=sin(g@f)`
`=sin(sinx^2)`
`f(ggf)=(ggf)^2`
`=[sin(sinx^2)]^2`
`=sin^2(sinx^2)`
`(sin(sinx^2))^2=sin(sinx^2)`
`sinx^2=y`
`(siy)^2=siny`
`(siny)^2-siny=0`
`siny[siny-1]=0`
`siny=0`
`y=npi`
`sinx^2=(4n+1)pi/2`
This is not possible
`siny=1`
`y=(4n+1)pi/2`
`sinx^2=npi`
`sinx^2=0`
`x^2=npi`
`x=pmsqrt(npi)`
`n in(0,1,2,3,4...)`.