Show that the evaluation map ∅_a of Example 18.10 satisfies the multiplicative requirement for a homomorphism.

Data from in18.10 Example 

Let F be the ring of all functions mapping IR. into IR. defined in Example 18.4. For each a ∈ R, we have the evaluation homomorphism ∅_a: F → R, where ∅_a(f) = f (a) for f ∈ F. We defined this homomorphism for the group (F, +) in Example 13.4, but we did not do much with it in group theory. We will be working a great deal with it in the rest of this text, for finding a real solution of a polynomial equation p(x) = 0 amounts precisely to finding a ∈ R such that ∅_a(p) = 0. Much of the remainder of this text deals with solving polynomial equations. We leave the demonstration of the multiplicative homomorphism property 2 for ∅_a to Exercise 35.