Is this a function? Is the set $\theta=\{\big((x,y),(3y,2x,x+y)\big):x,y ∈ \mathbb{R}\}$ a function? If so, what is its domain, codomain, and range?
This is probably a dumb question. I understand what a function is, but the three elements in the ordered pair got me confused.

A general definition of function follows that of relation, a subset of the cartesian product of two sets $A\times B$, with the further prescription that each element of $A$ is in relation with exactly one element of $B$.
In such view, your set $\theta$ seems a subset of $\mathbb{R}^2\times \mathbb{R}^3$.
The domain is $\mathbb{R}^2$ and the range or image is a subset of $\mathbb{R}^3$. When $\mathbb{R}^3$ is viewed as a linear space and given the explicit linear and homogeneous form of the given formulas, this subset is a subspace.
