Where can I look to investigate and learn about broader classes of symmetries for algebraic structures?

Question

Here's $f(x) = x^2,$ it has the property that $f(-x) = f(x).$

I have a couple questions related to this topic:

1.) Is there a broader name for symmetries of the form $f(h(x)) = f(x)$ for invertible $h$?

2.) Is there a broader name of symmetries that keep a manifold invariant, for the cases that a manifold is defined by a system of equations like $x^2 + y = z$, $z^2 - 1 = y^2$ and so on?

3.) How do we keep the "symmetry" invariant of the specific coordinates? Every parabola has a line of symmetry about its vertex, but in the example I provided, the line of symmetry is stuck at the line $x = 0,$ how do we classify and find symmetries that are irregardless of the specific coordinates, to say that a parabola has a symmetry about some line centered at its vertex?

4.) How do we know we've found all symmetries of a given manifold, at least in 2D or 3D cases?

Answer 1

1. In this generality they're just called "symmetries." You'll have to get more specific to get a specific answer to this.
2. It depends. Continuous symmetries are called homeomorphisms. Smooth symmetries are called diffeomorphisms. Generally, symmetries in some category are called automorphisms.
3. It depends. Generally we try to find a description of the object we're interested in that doesn't depend on a choice of coordinates. In general it's a hard question to find all symmetries of some object.
4. Again, in general it's a hard question to find all symmetries of some object. You know this the same way you know anything else in mathematics: by proving it. If you ask about a more specific example then more specific things can be said.