Find gof and fog when f: R → R and g: R → R is defined by

f(x) = x2 + 8 and g(x) = 3x3 + 1

Since, f:R → R and g:R → R


fog:R → R and gof:R → R


f(x)=x2 + 8 and g(x)=3x3 + 1


So, gof(x)= g(f(x))


gof(x)= g(x2 + 8)


gof(x)= 3(x2 + 8)3 + 1


⇒ gof(x)= 3(x6 + 512 + 24x4 + 192x2) + 1


⇒ gof(x)= 3x6 + 72x4 + 576x2 + 1537


Similarly, fog(x)=f(g(x))


⇒ fog(x)= f(3x3 + 1)


⇒ fog(x)=(3x3 + 1)2 + 8


⇒ fog(x)=(9x6 + 1 + 6x3) + 8


⇒ fog(x)=9x6 + 6x3 + 9


So, gof(x) = 3x6 + 72x4 + 576x2 + 1537 and fog(x) = 9x6 + 6x3 + 9


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