First, a "rule", by itself, is never a function. One way of defining a functions is, simply "a set of ordered pairs such that no pairs have the same first member with different second members". If we are given both a domain (the set of first members) and a "rule" (so that, for each first member we can calculate the second member) then we have a function.
Though it is not good notation, we are sometimes given a "rule" along with the (unstated!) assumption that the domain is the largest set of values to which the "rule" can be applied. That is, if we are given the rule "f(x)= (x-3)/(x- 3)", the default domain is "all x except 3". But it is still a function. Similarly, , is a function with domain "all non-negative x".
The most important of your examples is "y=5x^2 + x^1/4". If it were true that, as you say, "for x>0 [it] has two values", then it would NOT be a function. However, you are wrong that y, or x^1/4, "has two values". Yes, it is true that has two roots. But x^1/4 is, by definition, the positive root of y. The equatilon has (real number) solutions . The reason we need to write " " is that " ", by itself, means only one of those two solutions.
(in terms of complex numbers, there would be four solutions to .)