Double branch $\sqrt x$ or square function turned 90°?

Question

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.

Answer 1

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 $x$.

$$x=y^2$$ $$y=\pm\sqrt{x}$$

Answer 2

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.

Answer 3

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).

Answer 4

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 $x$.