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Odd And Even Functions


Date: 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/   
    
Associated Topics:
High School Functions

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