Question

Is the function `f(x)=(1/x^3+x)^5` even, odd or neither?

Answer 1

`f(x) = (1/x^3 + x)^5` is odd.

Explanation:

A function `f(x)` is even if and only if `f(-x) = f(x)`.
A function `f(x)` is odd if and only if `f(-x) = -f(x)`.

To check this function then, we will look at `f(-x)`.

`f(-x) = (1/(-x)^3 + (-x))^5 = (-1/x^3 -x)^5`

`=> f(-x) = ((-1)(1/x^3 + x))^5 = (-1)^5(1/x^3+x)^5`

`=> f(-x) = -(1/x^3 + x)^5 = -f(x)`

Thus in this case `f(x)` is odd.