I want to talk a little bit more about even and odd functions. Here’s a problem; suppose f(x) and g(x) are odd functions and h(x) is even, are the following even, odd or neither? Well I start with p of x equals f(x) times h(x). This is a product of the odd function f(x) and the even function h(x). I want to know whether the result’s going to be even or odd.
Well I can prove the result one way or the other by plugging in –x. Of course p(-x) is going to equals f(-x) times h(-x). Now since f is an odd function, f(-x) is –f(x). Since h is an even function, h (-x) is h of x. And this product is precisely p(x) so this is –p(x). We see that p is an odd function. Now what this tells us is, whenever you have the product of an odd and an even function, the result is an odd function.
Let’s take a look at another. Here we have an odd function divided by another odd function. F and g are odd, so what would the quotient be? Again we plug in –x to see what happens. We get f(-x) over g(-x). Now f(-x) because f is odd is –f(x). And g(-x) because g is odd is –g(x). The negative signs cancel leaving f(x) over g(x). Of course that’s just q(x). Back to out original function.
So it turns out that q is even. What this tells us is if you take a quotient of two odd functions, and make a new function out of it, the result will be an even function. Quotient of odd functions is actually even.