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Chapter 1: Problem 81

Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function.

Short Answer

Expert verified
The product of two odd functions and the product of two even functions both result in an even function.

Step by step solution

01

Prove that the product of two odd functions is an even function

Let's take \(f(x)\) and \(g(x)\) as two odd functions. This means that \(f(-x) = -f(x)\) and \(g(-x) = -g(x)\). Now, let \(h(x)\) be the product of the two functions, so \(h(x) = f(x)*g(x)\). Therefore, \(h(-x) = f(-x)*g(-x) = -f(x)*-g(x) = f(x)*g(x) = h(x)\). This satisfies the definition of an even function, hence the product of two odd functions is an even function.
02

Prove that the product of two even functions is an even function

Let's take \(f(x)\) and \(g(x)\) as two even functions. This means that \(f(-x) = f(x)\) and \(g(-x) = g(x)\). Now, let \(h(x)\) be the product of the two functions, so \(h(x) = f(x)*g(x)\). Therefore, \(h(-x) = f(-x)*g(-x) = f(x)*g(x) = h(x)\). This satisfies the definition of an even function, hence the product of two even functions is an even function.

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