How can a "non-function" be graphed? 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.

Any help on how to graph non-functions is appreciated. Thanks in advance.
 A: 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} $.
A: 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
A: 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.
