Question

How do you determine whether a function is odd, even, or neither: `h(x)= -x^3/(3x^2-9)`?

Answer 1

Odd

Explanation:

We substitude `x` with `-x` :

`h(-x)=-(-x)^3/(3(-x)^2-9)=-(-x^3)/(3x^2-9)=x^3/(3x^2-9)=`

`-h(x)`

Now because `h(-x)=-h(x)` the function is odd.

Answer 2

`h(x)` is an odd function.

Explanation:

We use the following condition:

`{ (f(-x)=f(x)), (f(-x)=-f(x)) :} => {: (f " is even"), (f " is odd") :}`

So for the given function:

`h(x) = -(x^3)/(3x^2-9)`

And so:

`h(-x) = -((-x)^3)/(3(-x)^2-9)`

`" " = -(-x^3)/(3x^2-9)`

`" " = (x^3)/(3x^2-9)`

`" " = -h(x)`

And we conclude that `h(x)` is an odd function.

We can verify this graphically, as odd functions have rotational symmetry about the origin:
graph{-(x^3)/(3x^2-9) [-20, 20, -10, 10]}