Mathematics Problem

y = x! (the factorial of x)

Note that factorial is defined for non-negative integers only.
(We do not consider the extension to gamma function at this moment.)

If the domain of x is all non-negative integers, that is the set {0, 1, 2, ...}, or its domain is any subset of this, then y = x! is a function of x.

This is because for every element in the domain, the value of x! is well-defined and uniquely determined.

If the domain of x is a larger set than {0, 1, 2, ...}, then y = x! is not a function of x.

For example, if the domain is the whole real line, then (-0.8)! is not defined.


2014-06-19 22:52:07 補充：
That means, it can be true or false.

It depends on the domain.

Actually, for this kind of question about function, you must state clearly all the details. (However, I understand that sometimes the question itself does not mention. There are many poor questions about function.)

True.

y is a function of x, that is to say, for each value of x (within its domain), there is a unique value for y.

The domain is, as mentioned by a previous post, non-negative integers (if ! excludes gamma functions).