Question

Prove that the product of two even (or two odd) functions is even.

Short answer

The product of two even functions or two odd functions is always an even function.

Step-by-step solution

Define the functions

Let's consider two even functions, \( f(x) \) and \( g(x) \), such that \( f(-x) = f(x) \) and \( g(-x) = g(x) \). We will prove this for even functions first, and the proof for odd functions will follow from the definition of odd functions.

Express the product function

The product of functions \( f(x) \) and \( g(x) \) is \( h(x) = f(x)g(x) \). Our goal is to prove that \( h(x) \) is also an even function.

Substitute and simplify

To show that \( h(x) \) is an even function, we need to demonstrate that \( h(-x) = h(x) \). So, we substitute -x into our product function: \( h(-x) = f(-x)g(-x) \). Since \( f(-x) = f(x) \) and \( g(-x) = g(x) \), then \( h(-x) = f(x)g(x) = h(x) \). Thus, the product of two even functions is also an even function.

Repeat for odd functions

For two odd functions \( p(x) \) and \( q(x) \), they satisfy \( p(-x) = -p(x) \) and \( q(-x) = -q(x) \). The product of these functions, \( r(x) = p(x)q(x) \), would yield \( r(-x) = -p(x)(-q(x)) = p(x)q(x) = r(x) \). Thus, the product of two odd functions is an even function also.