Help me understand this domain-related question (trig)

Factual Questions
1

Disclaimer: This is a question from an exam that I have already taken and has already been graded.

Here’s the question: Find the domain and range of g(x) = A cot(x-C) + D.

I put “all real numbers” for both and got domain wrong (range was right obviously). Taking the test, I thought hard about this. I reasoned that we don’t know what C is. Because the domain is defined as all the possible x values, I deduced therefore that x can be any numerical value.

The way I was looking at it, the graph g(x) = cot (x) has asymptotes at 0, pi, 2pi, etc. However, the graph of g(x) = cot(x+pi/2) has asymptotes at pi/2, 3pi/2, etc. Therefore, and this is the thrust of my argument, X can take on any value.

When I asked my professor, he told me that C is a parameter, not a variable. When I asked him for a value that X could not be, he said “C.” Now, I understand that I am wrong here, and other people wanted to talk to him so I just walked away. But I’m still not satisfied. I still believe that there is no real number that X could not be.

Can anyone shine some light on this for me?

2

When you are finding domain and range, g(x) = cot(x) isn’t the same as g(x) = cot(x+pi/2), though. Since C isn’t a variable, you can’t assume it changes to deal with whatever x is… I mean, cot(C-C) is undefined, isn’t it? I can’t see how you’d disagree with that.

3

Yeah, the question is better understood as: “For given values of A, C, and D, what are the domain and range of the function f(x)?” The answer would then be “all real numbers except those of the form C + n*pi, where n is an integer.”

4

Ah, that makes sense. C-C would of course be undefined. Thanks for the answers.