Question

How do you determine if `f(x) = x((x^2)-1)` is an even or odd function?

Answer 1

This function is an odd function.

Explanation:

To find out if the function `f(x)` is odd or even you have to calculate `f(-x)`.

If `f(-x)=f(x)` then the function is even,
else if `f(-x)=-f(x)` then the function is odd,
else we can say that the function is neither even nor odd.

In the example above we have:

`f(x)=x*(x^2-1)`

`f(-x)=(-x)*((-x)^2-1)=-x(x^2-1)`

So we see, that `f(-x)=f(x)`, which means that the function is odd.

We can also find if the function is odd or even looking at its graph.

If the function is even then Y-axis is the axis of symetry of its graph, else if the function is odd then the origin (0,0) is the graph's center of symetry.

graph{x*(x^2-1) [-10, 10, -5, 5]}