Question

How do you determine if `f(x) = x^3 + x sin^2 x` is an even or odd function?

Answer 1

`f` is odd.

Explanation:

A fun. `f` is `even` iff `AAx, f(-x)=f(x)` and `odd`, iff `f(-x)=-f(x)`.

For the given `f(x)=x^3+xsin^2x=x(x^2+sin^2x)`, we have, `f(-x)=-x*{(-x)^2+sin^2(-x)}=-x{x^2+(-sinx)^2}=-x*(x^2+sin^2x)=-f(x).`

Clearly, `f` is odd.