If $x$ and $y$ are independent variables and the functions $f_1(x,y)$ and $f_2(x,y)$ are defined as: $f_1(x,y) = x + y$ and $f_2(x,y) = 2(x + y)$, does it make sense to say that $f_1$ and $f_2$ are independent of each other?
I have 2 contradictory thoughts about it:
They are indeed independent because the explicit definitions of both of them are as functions of only the variables $x$ and $y$
They are not independent as one could express one in terms of the other, either $f_1 = f_2\,/\,2$ or $f_2 = 2\,f_1$.
If one lets $f:D \rightarrow \mathbb{R}$ and let $D \subseteq \mathbb{R}^n$ and $U \subseteq \mathbb{R}^m$. If $m=1$ then we say that $f$ is a scalar-valued function and if $m > 1$ we say that $f$ is a vector-values function.
If $n=1$ then we say that $f$ is a function of one variable, and if $n > 1$, we say that $f$ is a function of several variables. For each $x = (x_1, x_2, . . . , x_n) \in D$, the coordinates $x_i$ of $x$ are called the independent variables of $f$.
Take as an example $z=x^2+y^2$. The independent variables are $x, y$ the dependent variable is $z$.