Is $x^3+1$ considered even or an odd function?

Question

Question

Is $x^3+1$ considered even or an odd function?

I was wondering that because it shifted above by a factor of 1 so would that make it a non odd function?

Answer 1

$$f(x)=x^3+1$$

$$f(-1)=0$$

If it is an odd function, $f(1)=-f(-1)=0$ , which is not true.

If it is an even function $f(1)=f(-1)=0$, which is not true.

It is neither odd nor even.

Answer 2

You can see that neither $f(-x)=f(x)$ nor $f(-x)=-f(x)$ hold.

Answer 3

Even functions are those which have a mirror symmetry about $x=0$

Odd functions are those which exhibit rotational symmetry when rotated about $\pi$ radians about $(0,0)$

Yes, $x^3 - 1$ would have been odd if shifted by 1 to the left, but unfortunately that is not enough like @Siong Thye Goh pointed out.

Answer 4

Any arbitrary function $f(x)$ can be decomposed into an even and odd part:

$$f(x) = \underbrace{{\frac{1}{2}\left[f(x) + f(-x)\right]}}_{\text{Even part}} + \underbrace{\frac{1}{2}\left[f(x) - f(-x)\right]}_{\text{Odd part}}$$

In this case the even part is the constant function $1$, while the odd part is the function $x^3$.