Question

How do you determine if `f(x) = 3x^5 - 4x + 3` is an even or odd function?

Answer 1

`f(x)` is neither even nor odd.

Explanation:

`f(x) = 3x^5-4x+3`

For `f(x)` to be an even function it would have to satisfy `f(-x) = f(x)` for all `x`.

For `f(x)` to be an odd function it would have to satisfy `f(-x) = -f(x)` for all `x`.

With our example, we find `f(1) = 3-4+3 = 2` and `f(-1) = -3+4+3 = 4`

So neither condition is satisfied.

`color(white)()`
Actually, we could have spotted this another way:

For polynomial functions:
1. If all of the terms are of odd degree then the function is odd.
2. If all of the terms are of even degree then the function is even.
3. If some of the terms are of odd degree and some even then the function is neither odd nor even.

In our example, `3x^5` and `-4x` are of odd degree and `3` is of even (`0`) degree. So our `f(x)` is a mixture and neither odd nor even.