Definition of even functions for n dimensions

Question

Is there a generalisation of even functions for functions with multiple variables? If so, what are some concrete examples of the use of this definition?

Answer 1

Let $x\in \mathbb R^n$ be a vector and consider a function $f\colon \mathbb R^n\to \mathbb R$. Then $f$ is even if $f(x)=f(-x)$.

Here is an example. Let $f(x)=|y\cdot x|$ for any vector $y\in \mathbb R^n$. Then $f(x)=f(-x)$ by linearity (followed by taking absolute values).

(The same definition works in any module over any ring.)