Double branch $\sqrt x$ or square function turned 90°? I have this idea for a graph but don't know what function could describe it better.
The idea is something like the "squared" function turned $90$ degrees to the right, so that possible values for $x$ are always positive and $y$ may be both positive and negative.
The graphs of $\sqrt x$ and $-\sqrt x$ combined look good too, but I don't know how to write that as a single function (eh, I'm so bad with these things).
Basically, anything that may look like this will do.
Looking forward to some solution.
 A: What you're looking for can be described as a parabola opening towards the positive $x$ axis.  I'm going to refer to it as a "sideways parabola," since the "standard" parabola that people learn opens towards the positive $y$ axis.
The bad news: You cannot express a sideways parabola as a function of $x$.  Why?  Let's go back and look at a restriction on functions:

Vertical Line Test: For a relation to be a function, it must (colloquially) pass the vertical line test.  That is, you must be able to draw a vertical line anywhere on the relation's graph and the line must intersect the relation in at most one point.

In the picture below, I've marked the two intersections that a vertical line makes on a sideways parabola.  This shows that the sideways parabola is not a function.

However, the good news is that one may still describe such a graph with mathematical notation.  Two such ways are below, but keep in mind that they are not functions of $x$.
$$x=y^2$$
$$y=\pm\sqrt{x}$$
A: The graph you describe will have equation $x=y^2$.
You cannot write it in the form "$y=\langle\hbox{function of $x$}\rangle$" because, just as you pointed out, every $x>0$ will correspond to two $y$ values, not just one.
A: If you want the square function
$$ y = x^2 $$
turned 90 degrees to the right, you get the inverse
$$ x = y^2 $$
or written in another way (like you already mentioned)
$$ y = \pm \sqrt{x}. $$
I'm not actually sure what you mean by writing it as a single function as it has two branches (the positive and the negative one).
A: The curve obtained by combining the graphs of $y = \sqrt{x}$ and $y = -\sqrt{x}$ is $x = y^2$.  It is not a function of $x$ since there are two values of $y$ for each $x > 0$. However, you can use the parametric equations 
\begin{align*}
x(t) & = t^2\\
y(t) & = t
\end{align*}
to write both $x$ and $y$ as functions of a third variable $t$.  Observe that $x(t) = [y(t)]^2$.
By using parametric equations to express both $x$ and $y$ as functions of $t$, you can describe curves that are not necessarily functions of $x$.   
