# 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.

## 4 Answers

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 $x$.

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

• okay, somehow i didn't know about the vertical line. this is a really good explanation, thank you. – Don Jan 14 '15 at 4:22

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.

• Concise. Correct. +1. – MPW Jan 14 '15 at 4:55

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).

• what i meant comes from the fact that i'm stuck with this program which will only accept input like y = sqrt(x), so there's no +/-, and giving it sqrt(x)+(-sqrt(x)) makes no sense of course. – Don Jan 14 '15 at 4:20
• Well, in that case graphing a function with two branches like that seems to be impossible. – 655321 Jan 14 '15 at 4:30
• More accurately, $y = \sqrt{x}$ is the inverse of $y = x^2, x \geq 0$, while $y = -\sqrt{x}$ is the inverse of $y = x^2, x \leq 0$. The equation $x = y^2$ is not a function of $x$ since there are two values of $y$ for each value of $x > 0$. – N. F. Taussig Jan 14 '15 at 12:54

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 $x$.