Notation for defining $f:\Bbb{R}^n \to \Bbb{R}^m$?

Question

How to notate analytical defining of the function $f:\Bbb{R}^n \to \Bbb{R}^m$?

For example, for $n=3$, $m=2$, I notate it with m-tuples: $f(x_1, x_2, x_3) = (x_1+x_2, \frac{1}{x_2+x_3})$. Is this the correct nonation? Are there another ways fo notate it?

Answer 1

I'd write $\frac1{x_2+x_3}$ rather than $\frac1{x2+x3}$. Other than that, it's fine.

In some cases, it might be important to decide whether $\Bbb R^n$ and $\Bbb R^m$ are column or row vectors, which is why you might find notations like $$f(x_1,x_2,x_3)=\begin{pmatrix}x_2+x_3\\\frac1{x_2+x_3}\end{pmatrix}\\ f((x_1,x_2,x_3)^T)=\left(x_2+x_3,\frac1{x_2+x_3}\right)^T$$

However, this is typically left for the author to decide, because readability often outweights the need for disambiguation.