# Describing a function that has multiple inputs

I'm trying to find a way describe in mathematical terms a function that takes one or more inputs but the input and output of the function have the same dimensions or shape.

Let's say $x$ is a vector of length $n$. $y$ is also a vector of length $n$ is there a term that describes a function $f$ where $f(x,y) = z$ and $z$ has the same length as $x$ and $y$. The terms isomorphism and homomorphism have been thrown around but I feel like either these terms are not narrow enough or they are too complicated to understand in this simple case.

Essentially I'm looking for a term (and maybe a branch) that describes relations of this type, the shape of the input matches the shape of the output. And if there is a term that describes the opposite where the shape of the output does not match the shape of the input that would be helpful too!

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For your example, you could simply say that $f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a function from the cartesian product of $\mathbb{R}^n$ with itself to $\mathbb{R}^n$. This means exactly that $f$ takes pairs of vectors and outputs a single vector. If you also wanted this function to preserve the linear structure, you'd say that $f$ is an homomorphism or a linear transformation.
By the way, if the domain is the same as the codomain, e.g. $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$, we speak of an endofunction or endomorphism.