I had an idea that what if the Cartesian coordinate system is modified so that the origin is at $x=1$, the right side goes to infinity as usual, while the left side goes to 0 as opposed to negative infinity, so that the entire domain the $x$-axis can cover is $(0,\infty)$. I don't know if this qualifies as a coordinate system (or what its technical definition is, in fact I don't know much about maths; I just suddenly thought about this today), but I gave it a try. I thought of something like this, with $1$ at the center, $2,3,4,...$ on the right while $1/2, 1/3, 1/4, ...$ on the left.
I tried to sketch some functions in this system and I find it interesting. I concluded that to graph the function in this "coordinate system" I can graph the following piece-wise function with $f(x)$ being the original function:
$$f \left(\frac{1}{1-x} \right),\;x<0$$ $$f(x+1),\; x \geq0$$
And then pretend that the origin is $(1,0)$ and $-1$ on $x$-axis is actually $x=1/2$ while $1$ is actually $x=2$. The resulting curves are smooth (for example) if the functions themselves are differentiable, which are conspicuous but I nonetheless tried to prove with my very limited calculus knowledge that the graph is smooth at $y$-intercept since I did not immediately realize the conspicuous fact that I am merely stretching the interval $(0,1)$.
But I did find that this "coordinate system" might have some use as it is demonstrative of the behaviors of certain functions who have asymptotes at $0$. For the reciprocal functions, it visually shows their linear $(1/x)$, quadratic $(1/x^2)$, quartic $(1/x^3)$, etc.. properties (as displayed on the left of $y$-axis), which are easy to relate when thinking about this functions but are not so easy to see from just looking at the regular Cartesian graph. It is also good for visualizing the reciprocal relationships of $x^n$ and $x^{-n}$, which is also quite hard to see in normal Cartesian graph. It also visualizes the origin symmetry of logarithmic functions, which are easy to relate to when thinking about numerical log operations while hard to see from Cartesian graph.
Note: the shown $x$-axes in the pictures are invalid when viewing the graphs graphed in this "coordinate system" (if you may call it) and the $x$-axis should actually go like $1/4, 1/3, 1/2, 1, $ (origin), $2, 3, 4,...$
I realized this might actually be a thing so I was wondering is this kind of "coordinate system" or transformation documented anywhere and have a name or is it just too trivial?
No, this coordinate system has not before been studied in-depth, and so doesn't have a name. But it's not trivial as you say. That's not the right word. It's just very specific or particular.
I'm posting this CW answer so that users who confidently concur have something to vote on, and so this question doesn't stagnate in the Unanswered Questions Queue. If however anyone has an affirmative response to the question, please downvote this and post yours.