First, the boring answer. If $x$ and $y$ are (real-valued) variables, then $y(x)$ never makes sense. You cannot evaluate a real number at a real number. If $f$ is a variable of type 'function-whose-domain-is-the-reals', however, then $f(x)$ does make sense.
Of course, people abuse notation. e.g. they might name one of their real-number variables $y$, and then name one of their function-variables $y$. I don't like it, though. :(
Now for the more interesting answer. Dependent variables are more difficult to define since it's an issue of syntax and grammar -- things usually (and IMO unfortunately) glossed over at an introductory level.
Dependent variables tend to arise naturally when trying to express things. For example, in the course of solving a problem, I may introduce a variable $P$, whose domain is the unit circle. I will likely find it convenient to introduce other variables such as $x$ and $y$, and insist on an algebraic dependence: $P = (x,y)$. I may want other variables, such as $\theta$, with the obvious meaning intended.
Now, $P$ is just a point, and $x$, $y$, and $\theta$ are all just real numbers, and they have various interrelationships, such as $x^2 + y^2 = 1$. i.e. dependent variables. Dependent variables are often very useful for setting up problems and doing a variety of calculations.
However, they're not functions -- at least not in the usual sense. I can always consider a model -- a functions $\xi$ that transform the variables into specific numerical values that satisfy all of the dependencies; e.g. the function defined by $\xi("x") = 0$, $\xi("y") = 1$, $\xi("\theta") = \pi/2$, and $\xi("P") = (0,1)$. Of course, one often turns this on its head, and instead thinks of $\xi$ as living in some hypothetical state space, and turning the variables into functions: $x(\xi) = 0$, $y(\xi) = 1$, $\theta(\xi) = \pi/2$, and $P(\xi) = (0,1)$. Occasionally, you can conflate one of the variables with this "state", but it doesn't always work out nicely.
This idea is often needed for doing calculus. However, it is not needed for differential forms (i.e. $dx$, $dy$, $dP$, $d\theta$) -- $dx$ makes sense as a 'differential number' in the same way that $x$ makes sense as a "number". But that's another topic.
Anyways, relations among variables are all fine and dandy, but they're often awkward to work with -- in various situations functions are much easier to work with. So, I might be interested in writing down a bivariate function $f$ with the property:
If y > 0, then y = f(x)
And now that I have expressed the dependence of $x$ and $y$ on the domain $y>0$ in this fashion, I can use all of the nice properties I know about functions to solve problems. Similarly, I might do the same on the other half-circles: e.g. find another function $g$ such that
If x > 0, then x = g(y)
Since functions are much easier to work with, you usually see things first introduced in terms of functions. e.g. calculus is introduced as ways to work with functions. Of course, one usually work with dependent variables too, but it's learned through example -- and, unfortunately, often with the implicit suggestion that you're "supposed" to think of it as being short-hand for something involving functions. It's not until differential geometry do you get some rigorous foundations for doing things the way you would like with dependent variables. (of course, it often comes with the implicit understanding that they're really functions, such as the on 'state space' I described above, but meh)