Do $f(x) = ax_1 + bx_2 +c, \ x_1 \in R, x_2 \in R$ and $g(v) = wv + b, \ v \in R^2$ have the same domain? per mathworld

The term domain is most commonly used to describe the set of values
  for which a function (map, transformation, etc.) is defined. For
  example, a function f(x) that is defined for real values $x \in R$ has
  domain $\mathbb R$, and is sometimes said to be "a function over the reals."

Here are 2 functions.
$f(x) = ax_1 + bx_2 +c, \quad where \quad x_1 \in R, x_2 \in R, \quad \text{a, b, c are constants}$
$g(v) = wv + b, \quad where \quad v \in R^2, \quad \text{w, b are constants}$
Do they have the same domain, that is $R^2$? 
I guess so, and I need a double check.

You did NOT give two functions.
A function normaly consists of three things, where the domain and range are a part of it.
So a function is given by $f:D\to R$ with $x\mapsto f(x)$.
You just give:
$f(x)=ax_1+bx_2+c$. So you miss the first part, which is important.
What is odd here, is that there is no $x$ on the RHS of the functional equation. So I guess $(x_1,x_2)=x\in\mathbb{R}^2$ is meant.
Also this:

For example, a function $f(x)$ that is defined for real values x∈R has domain R, and is sometimes said to be "a function over the reals."

from your quote names the function $f(x)$, but the 'name' of that function is $f$.
$f(x)$ is a value in the range of $f$, which is a common mistake.
So do your 'functions' have the same domain?
They could. Nothing would stop you of defining them with $\mathbb{R}^2$ as domain.
But you could also define unique domains for each function. 
You can be really flexible here. What is for sure is just, that $D\subseteq\mathbb{R}^2$ in both cases for $f$ and $g$ to make sense.
