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# Symmetry of Graphs: Odd and Even Functions - Problem 3

###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

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