why is the product of two odd functions even?

Give an example.

To understand why the product of two odd functions is even, let's first clarify the definitions:

1. Odd function: A function f(x) is considered odd if f(-x) = -f(x) for all x in its domain. In other words, when you reflect the graph of an odd function across the y-axis, it remains unchanged.

2. Even function: A function g(x) is considered even if g(-x) = g(x) for all x in its domain. In other words, when you reflect the graph of an even function across the y-axis, it remains unchanged.

Now, let's consider the product of two odd functions, denoted as f(x) * g(x).

When we evaluate f(-x) * g(-x), we have:

f(-x) * g(-x) = -f(x) * -g(x)

Using the fact that -(-a) = a, we can simplify the above expression as:

f(-x) * g(-x) = f(x) * g(x)

By comparing this expression with the definition of an even function, we can conclude that the product of two odd functions, f(x) * g(x), is an even function.

In summary, two odd functions multiplied together result in an even function because the negative signs associated with each odd function cancel each other out when the functions are evaluated at negative x-values.