Why is it unacceptable to say "the range is a function of the domain"?
You are at the driving range, if you like golf. You have a bucket of balls that you hit into the field in front of you. Where the balls land (a "measurement", if you will) is a function of how you hit the balls. The domain of this function is the bucket of balls and the "range" is where each of the balls lands. In this case the range has zilch to do with the bucket of balls sitting in front of you, but only on how you hit the balls. Thus it is meaningless to say that where the balls land depends on the fact that you have a bucket of balls to begin with.
All the other answers are perfectly correct. Let me point out, however, that a common abuse of notation can lead to a sense in which the range is meaningfully a function of the domain!
Although the domain and range (well, codomain) are supposed to be made explicit when defining a function, it's quite common in natural language to just say e.g. $$\mbox{"let $f(x)=x^2$."}$$ Maybe the domain $\mathbb{R}$ is clear from context, but other domains - $\mathbb{C}$, $\mathbb{Z}$, all real numbers between $\pi$ and $17$ - would also make sense.
Arguably, when speaking quickly like this what we have is a kind of meta-function: if you tell me what domain your function $f(x)=x^2$ has, then I know what function you have in mind; but there's not an obvious "right" domain for it to have. So, in this sense, the range is a function of the domain. If the domain is $\mathbb{C}$, the range is $\mathbb{C}$; if the domain is $[0, 3]$, the range is $[0, 9]$; etc.
On the one hand, this is very silly, and is really only an issue because we were unclear in our language: we said something like "let $f(x)=x^2$" rather than "let $f(x)=x^2$ for $x\in\mathbb{R}$," etc. On the other hand, this is actually not silly at all: there are plenty of times we have a "definable" function with a variable "domain", in a very precise way. However, this happens much later on down the mathematical road. For now, you should not think of the range as a function of the domain.