I think I have a weak formal education in mathematics

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SUMMARY

This discussion centers on the understanding of mathematical concepts such as domain restrictions and the implications of equations. The user seeks clarity on the terminology related to domain and restrictions in equations, specifically in the context of the equations $$y^{2} = 25 - x^{2}$$ and $$xy = 1$$. The conversation highlights the importance of recognizing domain restrictions, such as $$-5 \leq x \leq 5$$ and $$x \neq 0$$, to ensure consistency in mathematical expressions. Additionally, the user requests recommendations for foundational mathematics books to strengthen their understanding.

PREREQUISITES
  • Understanding of basic algebraic equations
  • Familiarity with mathematical terminology such as domain and range
  • Knowledge of inequalities and their implications in functions
  • Basic graphing skills for visualizing equations
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  • Research the concept of domain in functions and equations
  • Study the implications of restrictions in algebraic expressions
  • Learn about graphing techniques for equations like $$y = 1/x$$
  • Explore recommended mathematics textbooks for foundational learning
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Students seeking to strengthen their mathematical foundation, educators looking for teaching resources, and anyone interested in clarifying concepts related to algebraic equations and their graphical representations.

xorg
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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?
 
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xorg said:
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.
xorg said:
Other example, the equation:
xorg said:
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.
xorg said:
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?
 

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