$x \in \mathbb{R}^2$ has $f(x)$ plotted on a $\mathbb{R}^3$ mesh like $(x,y,f(x,y))$. So is $f \in \mathbb{R}, \mathbb{R}^2$ or $\in \mathbb{R}^3$?

Question

$x \in \mathbb{R}^2$ has $f(x)$ plotted on a $\mathbb{R}^3$ mesh like $(x,y,f(x,y))$. So is $f \in \mathbb{R}, \mathbb{R}^2$ or $\in \mathbb{R}^3$?

Clearly $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. But in the plot visualization it would appear as if it's $f(x,y,z)=(f_x,f_y,f_z)$.

Answer 1

$f(x,y) \in \mathbb{R}$ but $(x,y,f(x,y)) \in \mathbb{R}^3$. It's just that.

It's like when plotting a function $f: \mathbb{R} \to \mathbb{R}$: the values of the function are in $\mathbb{R}$ but you need two variables to represent it (the independent one and the dependent one). In this case, you need three because you have two independent variables $(x,y)$ and one dependent variable $z = f(x,y)$.