calculate h(gf)(a)) and (hog)(f(a))

Question

It I based on dedekind's theorem of composition

Comments

Please provide definitions of these functions.

Answer 1

No definitions of the three functions have been provided, so an example will be used for illustration purposes.

f(x)=x+3, g(x)=x-3, h(x)=3x².

gf(x)=fg(x)=f(x)*g(x)=x²-9.

h୦g(x)=h(g(x))=h(x-3)=3(x-3)²=3x²-18x+27.

h(gf(a))=h(a²-9)=3(a²-9)²=3a⁴-54a²+243.

h୦g(f(a))=h୦g(a+3)=3(a+3)²-18(a+3)+27=

3a²+18a+27-18a-54+27=3a².