Question

Even and Odd Functions (a) Must the product of two even functions always be even? Give reasons for your answer. (b) Can anything be said about the product of two odd functions? Give reasons for your answer.

Short answer

(a) Yes, the product of two even functions is always even. (b) The product of two odd functions is even.

Step-by-step solution

Definition of Even and Odd Functions

An even function is any function \(f(x)\) that satisfies \(f(x) = f(-x)\) for all \(x\) in the function's domain. Meanwhile, an odd function is any function \(f(x)\) that satisfies \(f(x) = -f(-x)\) for all \(x\) in the function's domain.

Product of Two Even Functions

Let's denote two even functions as \(f(x)\) and \(g(x)\). Their product \(h(x) = f(x)g(x)\) should also be even if \(h(-x) = h(x)\). Substituting \(f(-x)\) and \(g(-x)\) for \(f(x)\) and \(g(x)\) respectively, because they are even, gives \(h(-x) = f(-x)g(-x) = f(x)g(x) = h(x)\). So, the product of two even functions is always even.

Product of Two Odd Functions

Now let's denote two odd functions as \(f(x)\) and \(g(x)\). Their product \(h(x) = f(x)g(x)\) should be odd if \(h(-x) = -h(x)\). Substituting \(-f(-x)\) and \(-g(-x)\) for \(f(x)\) and \(g(x)\) respectively, because they are odd, gives \(h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x)\). So, the product of two odd functions is actually even.