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I typed the equation $|x|=x$ on varius graphing calculators waiting to see the part
of the plane where $x\ge0$. I thought that because for every point $A(x,y)$ with $x\ge0$ the above equation is true. Instead only the y'y axis is displayed(as if the equation is only true for $x=0$).

I would like to draw this semiplane without using the inequality $x\ge0$ (which works) but an equation. Is there a problem or I don't understand something? Any help would be appreciated.

In fact I want to graph the semicirle $|x^2+y^2-1|+|x|-x=0$ but noticed that the problem is the $|x|-x=0$ part on every graphing calculator. Is there any other possible way?

John
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    You should add more specifics about the graphing calculator. In a sense, this is debugging the calculator or your input to it. You're correct that $|x|=x$ is the same as $x\geq 0$, so the reason that you don't get the same graph is unclear. It could be that the input is completely invalid. It could be that the calculator doesn't find all solutions to that equation. We simply don't know, so we can't proceed unless you are more specific as far as I can tell. – Matt Samuel Dec 28 '18 at 12:03
  • Many calculators don't know how to graph functions that not of the type $y=f(x)$ ... In this case it's easy to visualize the set of solutions, because every point on the right-hand side of the origin is a solution ... That is, if you're plotting on the $xy$-plane, for example. So I don't see a particular need to even graph the thing. It would just look like a plane where the right-hand side is black, and the left-hand side is white. – Matti P. Dec 28 '18 at 12:05
  • Desmos shows that it can't even compute a |x| function – For the love of maths Dec 28 '18 at 12:06
  • Apple’s built-in software Grapher handles these situations easily, in both two and three dimensions. – symplectomorphic Dec 28 '18 at 15:28

2 Answers2

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The problem is presumably that most graphing calculators work in two dimensions, i.e. they need an $x$ and a $y$ (sometimes also called $f(x)$). Each point on that plane is an ordered pair, $(x,y)$ (or $(x,f(x))$ if you're talking about the graph of a function in that plane).

Consider $|x|=x$. Where is your $y$? I suppose you could graph it in one dimension, in which case it would just be the ray $[0,\infty)$, but most graphing calculators don't care to do something so simple.

So, how do we find $|x|=x$ in a standard two-dimensional graph? Well, move everything to one side, define $f(x)$ by replacing $0$ with it, and then look at the graph: where $y=0$ is where the equality is satisfied.

$$|x| = x \;\;\; \text{becomes} \;\;\; x - |x| = 0 \;\;\; \text{becomes} \;\;\; f(x) = x - |x|$$

Graph of $f(x) = x - |x|$: the solutions to $|x|=x$ will be where $f(x)=0$:

enter image description here

PrincessEev
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  • Just because an equation has only $n$ variables doesn’t mean it can’t be graphed in $k>n$ dimensions. Consider the graph of the plane $x=0$ (with $n=1$) in $k=3$ dimensions. Yes, what you said is one way of looking the problem (and what you say about most graphing calculators is right). But it’s wrong to suggest the $y$ is “missing” instead of simply free (unconstrained). For what it’s worth, Apple’s built-in Grapher software graphs these things easily. – symplectomorphic Dec 28 '18 at 15:26
  • Which I touched on by saying you could graph it as a ray. But the average graphing calculator works in two dimensions or higher, and in that sense the $y$ is missing, i.e. the fact that a graphing calculator is being used is necessary. – PrincessEev Dec 28 '18 at 15:28
  • No, it’s not a ray: the graph of $|x|=x$ is a right half-plane in two dimensions. It’s a ray in one dimension. It’s a half-space in three dimensions, etc.... My point is that your answer doesn’t make it crystal clear which issues are limitations of common technology and which issues are the real mathematical ones. – symplectomorphic Dec 28 '18 at 15:31
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ok...from my aspect,

$1.$ where is your "y" variable? to plot any function you you must have to give a variable representing the function itself.you can write $y=x\text{ or }y=|x|$.But can't simply write $x=|x|$

$2.$ Now coming to your equation.I think you want to have a graph like $y=\sqrt{1-x^2}$ but your given function and this function is not a same thing.there is a problem in your function.that is, given, $$|x^2+y^2-1|+|x|-x=0$$ $$\implies |x^2+y^2-1|=x-|x|$$ $$\text{now this is only possible when $x\ge 0$.and every time for those values of $x$,$x-|x|$ yields $0$.}$$ you must have to add this condition while graphing it.because it is a necessary condition to the function be valid. so,we can say, $$|x^2+y^2-1|=0$$ now it is only possible when $(x^2+y^2-1)$ is $0$ itself.

Now look you can compute it in geogebra 3D .and that is your problem. https://www.geogebra.org/3d/ypk8vzc5

look where $z=0$

Rakibul Islam Prince
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  • Your first point is very confused. Functions aren’t the only things that have graphs; any equation in $n$ variables has a graph in $k\geq n$ dimensions (no matter whether the equation determines one of the variables as a function of the others): just plot the points satisfying the equation. The graph of the equation $|x|=x$ in two dimensions is the right half-plane. – symplectomorphic Dec 28 '18 at 15:30
  • I wanted to mean,if he wanted to plot it in desmos or in any graphing calculator then.. – Rakibul Islam Prince Dec 28 '18 at 15:52