Can a professor make your final pass or fail?
Why wouldn't it be allowed?


Legally? You didn't say what country you are in, but I'm not aware of any country that regulates courses at this level.
Your university's rules? You didn't say what university you are in, so it's impossible to say for sure. In general, though, the "instructor of record" has the authority to make decisions about how a course will be graded. 


Broadly speaking, this does not seem like a bizarre policy that some wacko professor implemented; I've actually seen this in other math departments. I guess the philosophy is that most entry-level math courses are about skills: they do not want to pass you onto more advanced courses unless you know these foundational skills well enough to do them in a measurable way. (I realize that you may not personally plan to take more advanced courses, but the principle is the same). 

Anyway: yes, unless your university has a policy that forbids it (e.g., a policy saying that all professors of a given course have to grade the same way), this policy is almost certainly allowed.