Determining whether a Function is Even or Odd

Date: 11/09/1999 at 00:06:41
From: Nick Lanning
Subject: Odd and Even Functions

How do you determine whether each function is an even function, an odd 
function, both or neither? The problems are as follows:

     y = x^5 - 4x

     y = 6x^3 - 3x + 5

     y = x^2 - 64

I am stuck on how to get started and what steps I should take. Can you 
help me?

Date: 11/09/1999 at 10:54:08
From: Doctor Rob
Subject: Re: Odd and Even Functions

Thanks for writing to Ask Dr. Math, Nick.

Test for evenness:

Substitute for (x,y) the pair (-x,y), and solve the two equations 
simultaneously. If the equations are dependent, the function is even.


Test for oddness:

Substitute for (x,y) the pair (-x,-y), and solve the two equations 
simultaneously. If the equations are dependent, the function is odd.


Example: Test whether y = x^5 - x is even.

     y = (-x)^5 - (-x),
     y = x^5 - x,

and, subtracting the first from the second,

   0 = 2*(x^5 - x),

which is false if x = 2, for example. Thus the function is not even.


Example: Test whether y = x^5 - x is odd.

     -y = (-x)^5 + (-x),
      y = x^5 - x,

and, adding these,

     0 = 0,

which is true for all values of x, which means the two equations are 
dependent, and the function is odd.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/