I think I have a weak formal education in mathematics

[xorg]

For example:
$$ y^{2} = 25- x^{2} $$
$$ y = \sqrt{25- x^{2}} , -5\leq x\leq 5 $$

This part: $$ , -5\leq x\leq 5 $$

What is the name of this?If it were a function, it would be the domain. And for equations, and resolutions of equations, what is the name?

Other example, the equation:
xy = 1

It would restriction X ∈ ℝ
However, if x ≠ 0 then
y =1/x, x ≠ 0

But then returning to the above equation,
xy = 1, It seems that it would have to have the same restriction x ≠ 0 and not X ∈ ℝ In order to have consistency.

What is the purpose of this topic?
I showed some examples of how I do not have a solid background. I'm not convinced of what I am doing. And I'm not sure how to improve it, but I really want.

Can anyone recommend books to form a solid foundation in mathematics, to feel peaceful with, for example, above questions?

 

[Mark44]

For example:
$$ y^{2} = 25- x^{2} $$
$$ y = \sqrt{25- x^{2}} , -5\leq x\leq 5 $$

This part: $$ , -5\leq x\leq 5 $$

What is the name of this?If it were a function, it would be the domain. And for equations, and resolutions of equations, what is the name?

The inequality indicates a restriction on the values of x. If you view y as being a function of x, the inequality explicitly gives the domain. Whether the inequality is present or not, the domain of the function is still $-5 \le x \le 5$.
Note that the second equation about is not equivalent to the first equation - their graphs are different.

Other example, the equation:

xy = 1

It would restriction X ∈ ℝ
However, if x ≠ 0 then
y =1/x, x ≠ 0

But then returning to the above equation,
xy = 1, It seems that it would have to have the same restriction x ≠ 0 and not X ∈ ℝ In order to have consistency.

The graphs of the equation xy = 1 and y = 1/x are identical. For the equation xy = 1 there is an implied restriction that x cannot equal 0.

What is the purpose of this topic?
I showed some examples of how I do not have a solid background. I'm not convinced of what I am doing. And I'm not sure how to improve it, but I really want.

Can anyone recommend books to form a solid foundation in mathematics, to feel peaceful with, for example, above questions?