how can a define that a function is neither even nor odd without using negation words or symbols?

Question

I understand all the definitons

If $f(-x) = f(x)$, the function is even.

If $f(-x) = - f(x)$, the function is odd.

If $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, the function is neither even nor odd.

now i dont know how to define that a function is neither even nor odd without using negation words or symbols

thanks for the help

Answer 1

If you have an even function, it should be symmetric about the y-axis. If you fold your graph across the y axis, both halves of the function should line up with each other.

If you have an odd function, you have to first flip the left half of the graph and then it will be symmetric when you reflect on the y axis.

That, or you can visualize rotating the entire right half of an odd function's graph by 180 degrees, and it should line up with the left half.

Answer 2

Even functions have a simmetry along the $x=0$ axis.

Odd functions have a simmetry along the $x=-y$ axis.

Functions that do not have either of this simmetries are neither even nor odd.

One example of such a function is $f(x)=x^2+x+1$ (plot).

Answer 3

if there exists a point x1 ∈ D such that f (−x1) is undefined

Answer 4

If you're allowed to use the "$>$" symbol then you could simulate inequality in the definition: $x \ne 0$ if and only if $x^2 > 0$. So, $f$ is neither even nor odd could be expressed as: $$ \left[\exists x \in \mathbb{R}, (f(x) - f(-x))^2 > 0 \right] \wedge \left[\exists y \in \mathbb{R}, (f(y) + f(-y))^2 > 0 \right]. $$

(However, as the other commenters have indicated, this is still somewhat artificial.)