Odd And Even FunctionsDate: 01/11/98 at 16:46:00 From: Nicole Subject: Odd and Even Functions I need to know the differences between odd and even functions and examples of each. I've tried looking them up in my Trig book but it's too confusing! Thanks, Nicole Date: 02/24/98 at 17:28:50 From: Doctor Sonya Subject: Re: Odd and Even Functions Dear Nicole, The easiest differences to understand between even and odd functions are graphical. I'll start with even functions, and tell you what I mean. A function f(x) is even if f(x) = f(-x). Some examples of even functions are f(x) = x^2 (x^2 means "x squared") f(x) = x^4 + 3 f(x) = cos(x) Check to see that all of these fit the definition. If we have a function where f(x) = f(-x), what does this say about the graph? It means that when we plug in 2 and -2, we'll get the same output. Therefore pairs of points (2, -2), (3, -3), (7.6558, -7.6558), etc. on either side of the y-axis are always at the same height. It would probably help to draw some of the examples of even functions so you can see what I was talking about. You'll notice that each of these examples has a kind of symmetry to it. In fact, the part of the function to the right of the y-axis should be exactly the same as the part to the left of the y-axis. Make sure you understand why this is so. In math terms, even functions are said to be "reflexive about the y-axis". If a function g(x) is odd, it has the property that: g(x) = -g(-x) Some examples of odd functions are: g(x) = x^3 g(x) = sinx g(x) = x^3 + x Odd functions also have a kind of symmetry to them. I'll leave it up to you to figure out what it is. Look at the graphs and see what you come up with. Also, there's a very famous theorem about even and odd functions: Every function can be written as the sum of an even function and an odd one. -Doctor Sonya, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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