I'll take sine as an example for this question.
I've always taken $ \arcsin(x) $ as a single-valued function, however reading Wolfram MathWorld's page on "Inverse Sine" (http://mathworld.wolfram.com/InverseSine.html) I've found that it is actually multi-valued, and $ \operatorname{Arcsin}(x) $ is typically how we write the single-valued version of it.
My question is: is it more correct to treat $ \arcsin(x) $ as a set of values, so that if we have $ y = \sin(x) $, then $ x \in \arcsin(x) $, rather than $ x = \arcsin(x) $?
And, as a side question, is there a reason why Wolfram Alpha's plot of $ y = \arcsin(x) $ treats the function as single-valued, despite Wolfram MathWorld's statement that it is multi-valued?