Consider f: N → N, g: N → N and h: N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.
We have,
ho(gof)(x)=h(gof(x))=h(g(f(x)))
= h(g(2x)) = h(3(2x) + 4)
= h(6x + 4) = sin(6x + 4) ∀x ∈N
((hog)of)(x) = (hog)(f(x))= (hog)(2x)
=h(g(2x))=h(3(2x) + 4)
=h(6x + 4) = sin(6x + 4) ∀x ∈N
This shows, ho(gof) = (hog)of