Does $f(X)$ with $X=[x_1, x_2]$ mean $X$ appear as a whole in the definition of $f$? Let $f$ be a mapping with $f(X)$ and $X=[x_1, x_2] \in \mathbb R^2$.
So does $f(X) = g(x_1, x_2)$ make sense for any mapping $g$ with the same range as $f$?
Must $X$ be able to appear as a whole in the representation of $f(X)$? For example, in $f(X) = x_1 + x_2 = X^T [1,1]^T$, $X$ can be represented as a whole. Another example, in $f(X) = x_1 \times x_2$, $X$ seems not be able to be represented as a whole. 
If $f(X)$ doesn't require $X$ to be able appear as a whole, how can we wrote  a notation that requires $X$ to appear as a whole in $f(X)$?
Thanks in advance!
 A: 
Does $f(X) = g(x_1,x_2)$ make sense for any mapping $g$ with the same range as $f$?

Yes, it does. Read on for an explanation why.

Must $X$ be able to appear as a whole in the representation of $f(X)$?

Again, yes, it does. Fortunately, this is always possible. Namely, the defining property of $X = [x_1, x_2]$, that justifies calling it a "tuple" to begin with, is that $x_1$ and $x_2$ together uniquely determine $X$, and vice versa, that $X$ uniquely fixes $x_1$ and $x_2$.
Let us immediately pass to the general case of an $n$-tuple $X \in S^d$ for some set $S$.
Usually, when we want to use the coordinates of $X$, we introduce the so-called projection functions (often simply called projections):
$$p_i: S^d \to S: p_i (X) = x_i$$
where $x_i$ is in the $i$th position of the tuple $X$.
Using these $p_i$, we can e.g. rewrite $g(x_1, x_2)$ as:
$$g(x_1, x_2) = g(p_1 (X), p_2 (X))$$
in which $X$ only occurs "as a whole".
Now, once we have been comforted ourselves with the use of the $p_i$, it is very convenient to start writing $x_i$ instead of $p_i (X)$ when we define functions on tuples. In the end, this is just a small abuse of notation that greatly simplifies formulas, but does not compromise the full rigour in terms of the $p_i$ lying underneath.
