Operations such as $\sqrt{f(x)^2}$, $f(x) + a - a$, are obvious candidates for such a set. However, e.g., for the line $y = -x$, it seems to me to be non-trivial that $x^3 + y^3 = 0$ will plot the same line but $x^2 + y^2 = 0$ won't. Translation between coordinate systems also seems to be a non-trivial example. Is there any way to designate such a set? (Could this be generalized to other types of curves?)
The following are some more thoughts on the question:
It would be interesting in order to find alternate equation forms that might make more clear certain properties of a curve. For instance, $\frac{x}{a} + \frac{x}{b} = 1$ makes immediately obvious the abscissa and ordinate at origin. But we know that under some types of algebra, $ax + by + c = 0$ might fail to be represented by $\frac{x}{a} + \frac{x}{b} = 1$. So we're lead to think that these two equations plot a line by virtue of legitimate operations between them.
The equation of a plane also seems to be nicely related to the general form of a line, if $r_0 = (x_0, y_0)$ and $r = (x, y)$ are two vectors pointing to the plane and the normal is $n = (n_x, n_y)$. If $\circ$ between vectors is the dot product, $(x - x_0, y - y_0) \circ n = (x-x_0)*n_x + (y - y_0)*n_y = n_x*x + n_y * y - (x_0n_x + y_0n_y) = a*x + b*y + c = 0$
The idea is to be able to see how the form of an equation can be altered, not the content of the variables. It seems odd to me that very complicated equations could have the same plotted curve as simple forms, but that this property wouldn't appear by virtue of the equation themselves, or the set of valid operations on this equation. This might seem weird, but say it is never immediately obvious that $ax + by + c = 0$ plots a line, or $x^2 + y^2 = r^2$ plots a circle, unless we actually do the plotting, and $ax + by + c = 0$ seems way less fundamental than $y = mx + b$.
Note that in the case of a circle, we have the pythagorean theorem that seems to be its clearest representation with the methods of analytic geometry, and the moment an equation can be said to share some sort of operation set with the pythagorean theorem, we know we're speaking of a circle. It seems that if we could somehow draw the operation set of a circle, we would get something like the pythagorean theorem, and that this operation set gets somehow deformed in order to give a representation onto the cartesian plane. For a translated circle with center $(h, k)$, $x^2 - 2xh + h^2 + y^2 - 2yk + k^2 = r^2$ means absolutely nothing to us, but the form $(x-h)^2 + (y-k)^2 = r^2$ is clear as day.
(Sorry if I am being unclear, I am doing my best to properly expose the question)
This is not a full answer, since the question is somewhat vague, although I know what you are trying to get at. I'll try my best.
From a somewhat algebraic-geometric perspective: If you take any function that vanishes nowhere outside your line, by which I mean a function $g$ for which $g(x,y) = 0$ implies $ax + by + c = 0$, then the set of vanishings of $fg$ is also the line. In a special case $f^n$ for any $1 \leq n \in \mathbb N$ works.
Or in general: For any function $f : X \to \mathbb R$, define $V(f) = \{ x \in X : f(x) = 0 \} = f^{-1}(\{0\})$. For any two functions $f$ and $g$, we have $V(fg) = V(f) \cup V(g)$. So if $V(g) \subseteq V(f)$, then $V(fg) = V(f)$.
So this is one example of what we can do to your function that doesn't change its zero set. But I know this is not what you are looking for based on your elaborated comments. You are looking for ways to alter your function to immediately extract more information from it. I want to say that unfortunately we can't do this in general.
Many curves (or spaces in general) defined by seemingly simple equations can be extremely complicated. For example the elliptic curves $y^2=x^3+ax+b$ has been an active area of research for geometry, number theory and cryptography. The structures of these curves are complicated enough that they are used to encode cryptographic information, here is a nice intro to read about it.
In general we can't easily extract too much information about the geometric structure defined by a equation by just rewriting or altering the equation. And I think this is a good thing. A lot of equations have deep significance in the studies of algebra and number theory, by examining their geometric complications we can be rewarded with more insights about other fields in mathematics.