Question

How do you determine if `h(x)= x^7+x^3+7` is an even or odd function?

Answer 1

It is neither.

Explanation:

The shortcut method is to note that `h(x)` is a polynomial with terms with a mixture of odd degree (`x^7` and `x^3`) and even degree (`7`), so it is neither an odd nor even function.

• An even function satisfies `f(-x) = f(x)` for all `x` in the domain.

• An odd function satisfies `f(-x) = -f(x)` for all `x` in the domain.

We find:

`h(1) = 1+1+7=9`

`h(-1) = -1-1+7 = 5`

So `h(x)` satisfies neither condition.

In general, one effective method to tell whether a function `f(x)` is odd, even or neither is to substitute `-x` for `x` into its definition and see if the result simplifies to `f(x)` or `-f(x)`.