Associated Topics || Dr. Math Home || Search Dr. Math

### 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,

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/
```
Associated Topics:
High School Functions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search