While trying to make a function to be put into a computer shader, i've come up with an expression in the following form: $$f(x) = \sum_{i} \frac{1}{x - D_i}$$
But the thing is that i actually need the inverse of that function for it to do what i want
But i have not enough math knowledge to actually come up with that function inverse, if there even exists one
Obs: D is an arbitrary list of numbers
How would i come up with the inverse of that function, or if doesn't exists, why?
Let's look at $$ f(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-10} $$ and let's say that $x = 1, 3, 10$ are the "special" points.
If we pick an $x$ that's just a tiny bit greater than one of the special points, say $x = 3.00001$, then the first term will be about $1/2$l the third will be about $1/(-7)$, and the middle one will be...about $10,000$. By adjusting $x$ a tiny bit, you could get any number from $9000$ to $11000$, although finding exactly the right $x$ could be tricky because of the other terms and their slowly-changing values as you diddle with $x$.
You could do exactly the same thing with $x = 10.00001$, once again producing any output between $9000$ and $11000$.
But that shows you that this function (or any other function that looks like it) is not one-to-one, i.e., there are two different $x$ values that produce the same $y$-value. So it does not have an inverse.
Sorry to disappoint you, but that's the way it is.
(By the way, if you use Desmos to plot the function I described, it'll be visually clear that almost any horizontal line intersects the graph at multiple locations.)