Question

How do you know if `f(x)=x^3+1` is an even or odd function?

Answer 1

Odd.

Explanation:

Quite simply, whether a function is even or odd is whether the degree (the largest exponent) is even or odd.
The degree of `x^3 + 1` is `3`, which is odd, making this an odd function.

Answer 2

Neither.

Explanation:

An function is even if: `f(-x)=f(x)`

A function is odd if: `f(-x)=-f(x)`

If `f(-x)=x^3+1`, the function is even.
If `f(-x)=-x^3-1`, the function is odd.

So, find `f(-x)`.

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

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

This function is neither odd nor even.

A good way to check is by recognizing that even functions are reflections of themselves over the `y`-axis and odd functions are reflections of themselves over the `x`-axis.

This is the graph of `x^3+1`: graph{x^3+1 [-10, 10, -5, 5]}

Notice that if the graph were shifted down one unit (the function `x^3`), the graph would be a reflection of itself over the `x`-axis and would indeed be odd.