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.