Notation of 2 variables function I often see 2 versions of notation and it really confuses me.
You can write $f(x,y)=x^2+y^2$ and this is of course a function of 2 variables
But you can also write $z=x^2+y^2$,and this is also a function of 2 variables because z is the corresponding constant value
So is it correct to simply say that $z=f(x,y)$?

The notation $f(x,y)=x^2+y^2$ (assuming that the domain is $\Bbb R^2$) describes a function from $\Bbb R^2$ to $\Bbb R$, which, formally, is a subset of $\Bbb R^3$. The notation $z=x^2+y^2$ describes a subset of $\Bbb R^3$, namely the elements that hold the equation, so they are really the same object. The precisest ways to write them are:

*

*As a function:
$$\begin{array}{c}f:\Bbb R^2\to \Bbb R\\(x,y)\mapsto x^2+y^2\end{array}$$

*As a set:
$$\{(x,y,z)\in \Bbb R^3:z=x^2+y^2\}$$
(I have assumed that $\Bbb R^2\times \Bbb R=\Bbb R^3$, which, strictly speaking, is not true, but I don't know any situation where the difference is relevant).
