31) If f(x) = x², g(x) = 3x and h(x) = x - 2, Prove that (f∘g)∘h = f∘(g∘h).
Answer:
LHS: (f∘g)∘h
f∘g = f[g(x)] = f(3x) = (3x)² = 9x²
(f∘g)∘h = (f∘g)[h(x)] = (f∘g)[x - 2] = 9(x - 2)²
RHS: f∘(g∘h)
g∘h = g[h(x)] = g[x - 2] = 3(x - 2)
f∘(g∘h) = f[g(h(x))] = f[3(x - 2)] = [3(x - 2)]² = 9(x - 2)²
Since LHS = RHS, (f∘g)∘h = f∘(g∘h) is proved.