Show that the multiplication defined on the set F of functions in Example 18.4 satisfies axioms R2 and R3 for a ring.

Data from  18.4 Example 

Let F be the set of all functions f: R → R We know that (F, +) is an abelian group under the usual function addition,  (f + g)(x) = f(x) + g(x). 

We define multiplication on F by (fg)(x) = f(x)g(x).

That is,fg is the function whose value at xis f(x)g(x). It is readily checked that Fis a ring; we leave the demonstration to Exercise 34. We have used this juxtaposition notation aµ for the composite function a(µ(x)) when discussing permutation multiplication. If we were to use both function multiplication and function composition in F, we would use the notation f o g for the composite function. However, we will be using composition of functions almost exclusively with homomorphisms, which we will denote by Greek letters, and the usual product defined in this example chiefly when multiplying polynomial function f (x)g(x), so no confusion should result.