Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The statement is correct. The product of two odd functions results in an even function, and the product of two even functions also results in an even function, as proven by the properties of even and odd functions

Step 1: Proving the product of two odd functions is an even function

Let us take two odd functions \(f(x)\) and \(g(x)\) such that \(f(x) = -f(-x)\) and \(g(x) = -g(-x)\). The product \(h(x) = f(x)g(x)\) is also a function. Substitute \(x\) with \(-x\) in the function \(h(x)\), obtaining \(h(-x) = f(-x)g(-x)\). Now, replace \(f(-x)\) and \(g(-x)\) with their odd function properties, yielding \(h(-x) = -f(x)(-g(x)) = f(x)g(x) = h(x)\). Therefore \(h(x) = h(-x)\), which means \(h(x) = f(x)g(x)\) is an even function.

Step 2: Proving the product of two even functions is an even function

Consider two even functions \(p(x)\) and \(q(x)\), such that \(p(x) = p(-x)\) and \(q(x) = q(-x)\). The product \(r(x) = p(x)q(x)\) is a function. Replace \(x\) with \(-x\) in \(r(x)\), yielding \(r(-x) = p(-x)q(-x)\). Now, substitute \(p(-x)\) and \(q(-x)\) with their even function properties, yielding \(r(-x) = p(x)q(x) = r(x)\). Therefore \(r(x) = r(-x)\), which indicates that \(r(x) = p(x)q(x)\) is an even function.