2

I am trying to create a program that requires the graphing of a "non-function", i.e, a function where "x" isn't strictly dependent on "y".

Let's answer the obvious question: Is it even possible?

Yes, it is possible, as demonstrated by online graphing calculators such as Desmos and GeoGebra, as well as several other open-source softwares. However, I haven't had success in figuring out how these graphing calculators graph non-functions. I have done some research, and have found nothing helpful.

Now, I'm looking for means to graph any non-functions, not just simple ones such as a circle or a sideways parabola, which simply require the graphs of +f(x) and -f(x). Maybe more complicated ones such as sin(x) + sin(y)= 1. The following graph is of the said non-function, as graphed by Desmos.

sin(x)+sin(y)=1

Any help on how to graph non-functions is appreciated. Thanks in advance.

Rapid Readers
  • 175

3 Answers3

1

A lot of the time examples like these "look" like functions locally. Therefore they can broken up into little pieces by using a valid set of initial conditions or a function which is valid and describes part of the overall relation you are trying to graph.

In your example we can convert the equation to look like:

$$ y = sin^{-1}(1-sin(x)) $$

Which gives a starting point looking like the bottom half of a single warped sphere. Then we see that since $sin(y)=sin(\pi-y)$ we can cap the warped sphere. Then we can argue that the equation is clearly unchanged if we were to replace $(x,y)$ with $(x+2\pi n ,y+2\pi m)$ (for integer values of m,n) and so we can repeat our pattern throughout the entire grid due to the periodicity of the $sin$ function.

Another good example is $y^2 = x$ where we have two sets of solutions corresponding to $y=\pm \sqrt{x} $.

Ryan Watt
  • 36
  • I really like your explanation. However, there is a problem. Some non-functions cannot be simplified exclusively to have y be dependent on x. For instance, x arcsin( x/y) + sqrt( y^2 - x^2) = 4xy, which isn't even symmetric like the other examples, but can still be graphed by Desmos. – Rapid Readers Sep 18 '17 at 04:11
  • If you are alright with approximate solutions then you can differentiate w.r.t. x to isolate for $\frac{dy}{dx}$ and then expand around as many sets of valid initial conditions as you can find. I'm sure there are some fancier tricks for finding a complete graph for an equation but I was just trying to point out that such an equation acts like a function somewhere around every set of valid initial conditions. – Ryan Watt Sep 19 '17 at 06:36
0

An aproach for this would be to take some values.
x arcsin( x/y) + sqrt( y^2 - x^2) = 4xy
For example x=5. We substitute this value. And later we try different values for y until we have a equality (left part is some as right side). Some mechanisms help to do this, looking on how quick we get a approximation. I would first convert this to a equation equal to 0 :
x arcsin( x/y) + sqrt( y^2 - x^2) - 4xy = 0
This is trivial, and now for each x-value, we can get a y value (or many y-values) using a aproximation methode like Newton, which use derivates.
Daniel

P.S. see https://en.wikipedia.org/wiki/Newton%27s_method

Daniel Pol
  • 350
0

According from what you asked, let's see images below and take conclusions.

https://i.ytimg.com/vi/y7N0Pr3N1RY/maxresdefault.jpg (points tracing)

https://i.stack.imgur.com/SAubS.png (from the same equation in 3D graph)

To trace a XY graph which is non-function, it is actually calculate the equation with every X and Y in the graph in range. And it is determined by intersection between XY plane.

A-KouZ1
  • 1