Question

Is the function `f(x) = x^3 + x sin^2 x` even, odd or neither?

Answer 1

`f(x)` is odd

Explanation:

A function is even if if exhibits the property `f(-x) = f(x)`
A function is odd if it exhibits the property `f(-x) = -f(x)`

Let check for `f(x)`:
`f(-x)`
`= (-x)^3 + (-x) sin^2 (-x)`
`=-x^3-xsin^2 x`
`=-f(x)`

Thus, `f(x)` is odd. You can confirm this by graphing. Since `x^3` is odd and `xsin^2 x` is odd, therefore `f(x)=x^3 + xsin^2 x` is odd.

graph{x*(sin(x))^2 [-10.32, 10.295, -5.155, 5.155]}