Question

How do you know if `f(x) =-x^3 + 6x` is an even or odd function?

Answer 1

Odd.

Explanation:

The parity of a function can be found through finding `f(-x)`:

• If `f(-x)=f(x)`, then the function is even.
• If `f(-x)=-f(x)`, then the function is odd.

Here,

`f(-x)=-(-x)^3+6(-x)`

`f(-x)=-(-(x^3))-6x`

`f(-x)=x^3-6x`

`f(-x)=-(-x^3+6x)`

`f(-x)=-f(x)`

Thus, the function is odd. Notice how the function is comprised totally of odd-powered variables.

Odd functions have the special property of being "origin symmetrical", meaning it's a reflection of itself over the `x` and `y` axes.

graph{-x^3+6x [-16.02, 16.02, -8.01, 8.01]}