# Odd and Even Functions

Function y=f(x) is called even if for any x from domain of f we have that f(-x)=f(x).

For example, y=x^2 is even function because f(-x)=(-x)^2=x^2=f(x). Another examples of even functions are y=x^4 and y=x^6.

Function y=f(x) is called odd if for any x from domain of f we have that f(-x)=-f(x).

For example, y=x^3 is odd function because f(-x)=(-x)^3=-(x)^3=-f(x). Another examples of odd functions are y=x^5 and y=x^7.

If for at least one pair of values x and -x f(-x)!=f(x) and for at least one pair of values f(-x)!=-f(x), then function is neither even, nor odd.

Domain X of even and odd functions should have following propery: if x in X then -x in X, i.e. X is symmetric (with respect to origin) set.

Example 1. Determine whether the following function is even or odd: y=x^(20).

We have that f(x)=x^(20), f(-x)=(-x)^(20)=x^(20). Therefore, f(-x)=f(x), so function is even.

Example 2. Determine whether the following function is even or odd: y=x^(13).

We have that f(x)=x^(13), f(-x)=(-x)^(13)=-x^(13). Therefore, f(-x)=-f(x), so function is odd.

Example 3. Determine whether the following function is even or odd: y=(x-4)/(x^2-9).

We have that f(x)=(x-4)/(x^2-9), f(-x)=(-x-4)/((-x)^2-9)=-(x+4)/(x^2+9). Since, f(-x)!=f(x) and f(-x)!=-f(x), function is neither even, nor odd.