Even Odd Function

If `f(x)=f(-x)` for every `x` in the domain of `f` then f is an even function.

For example, `f(x)=x^2` is even because for every `x` `f(-x)=(-x)^2=x^2=f(x)`.

If `f(-x)=-f(x)` for every `x` in the domain of `f` then `f` is an odd function.

For example, `f(x)=x^5` is odd because for every `x` `f(-x)=(-x)^5=-x^5=-f(x)`.

Consider graph of the two functions: `y=x^2` and `y=1/4 x^3`.

even and odd functions

Since `y=x^2` is even function then we draw it on interval `(0,oo)` (red solid line) and then reflect it about y-axis (red dashed line)

Since `y=1/4 x^3` is odd function then we draw it on interval `(0,oo)` (purple solid line) and then reflect it about origin (purple dashed line). In other words we rotate graph `180^0` counterclockwise.

Example 1. Determine whether function `y=1/x^2` is even or odd.

Since `f(-x)=1/((-x)^2)=1/x^2=f(x)` then function is even.

Example 2. Determine whether function `y=x^3-x` is even or odd.

Since `f(-x)=(-x)^3-(-x)=x^3+x=-(x^3-x)=-f(x)` then function is odd.

Example 3. Determine whether function `y=x^3+x^2` is even or odd.

Since `f(-x)=(-x)^3+(-x)^2=-x^3+x^2` then `f(-x)!=f(x)` and `f(-x)!=-f(x)`. This means that function is neither even, nor odd.