What does it mean for a function to be oddly and evenly symmetric?

Question

From understanding, a function describes a relationship between multiple variables, and has unique values across all possible values on one axis while not duplicating violating a vertical line test to determine if values get repeated, and can also potentially be symmetric.

Symmetry, from what I understand, is a property of a function that exists in two types, either even or odd. Both seem clear in my mind to indicate information preservation of that function after that function is transformed from that Quadrant alone into the possible quadrants that do not cross into a quadrant where the remainder of the function exists.

If a function has symmetry about the x axis we reflect off the y axis check if there is a match in corresponding values then we get an even match on both sides then the function seems clear in my mind to be symmetric.

For odd, my reasoning, without looking for the formulaic definition is to construct a orthogonal line, which I am assuming as at a 45 degree from the x, y axis. If this function is rotated from Quadrant 1 to the Quadrant 3 (180 degrees), as an example using the newly constructed orthogonal axis, then it becomes clear to me that the function is becomes symmetric after that rotation but across the orthogonal diagonal.

What characteristic other than observation of sign flip across the x, y, values, makes it so that a name such as odd is designated to describe this type of symmetry, and also does that characteristic apply to even functions too? Is there a numeric, or graphical reason which I am not seeing directly that can simply explain the reasoning being the two names, even ( number divisible by 2 ) and another odd ( not divisible into an whole number by another even number )?

Answer 1

Even symmetry: $f(-x)=f(x)$, e.g. $f(x)=\cos x$ or $f(x)=x^2$.

The graph is symmetric about the $y$-axis.

Odd symmetry: $f(-x)=-f(x)$, e.g. $f(x)=\sin x$ or $f(x)=x^3$

The graph is symmetric with the reflection in first the $y$- then the $x$-axis.

Answer 2

The most basic examples of even functions are $f(x)=x^n$ with even $n$, the most basic examples of even functions are $f(x)=x^n$ with odd $n$. Note that for even $n$, we have $(-x)^n=x^n$, so the graph is symmetric under reflection at the $y$-axis, and for odd $n$, we have $(-x)^n=-x^n$, so we have to reflect both at the $y$ and the $x$ axis (or equivalently: rotate by $180^\circ$ around the origin, aka. a point reflection at the origin).

A general function is called even/odd if it has one of these symmetries just like the even/odd powers.