Why is it unacceptable to say "the range is a function of the domain"? I understand that a function is defined as a correspondence between two sets, the domain and the range.
While the definitions of the domain of  a function and the range of a function are said to be contained within the definition of a function.
Now, when a measurement varies over time we say that it is a function of time. Why is it not generally acceptable to replace the word "measurement" with range and the word "time" with domain, and say that the range is a function of the domain, the same way as we say y is a function of x?

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.
