Demystify College Algebra with Step-by-Step Examples and Expert Guidance

Given f(x) = absoulte value of X and g(x)= x-2. Find $(f+g)(x),(f-<!--\; -\; -->g)(x),(fg)(x),(f/g)(x)$ and $(f\circ <!--\; \circ \; -->g)(x)$

Prove the identity
$\csc <!--\; \; -->x-<!--\; -\; -->\cot <!--\; \; -->x\cdot <!--\; \cdot \; -->\cos <!--\; \; -->x=\sin <!--\; \; -->x$

I am from a non-mathematics background but the course that I am taking in probability class is built on measure theory. I found many theorems stated in the book hard to comprehend and I think that's partly due to my insufficient maths background. I am self-learning real analysis currently and think it may be equally important to systematically learn abstract algebra if I really want to know the logic behind the probability class I am currently taking.
I find myself completely lost proceeding to the following section of the text.

"Let $R$ be a relation, $A$ a mathematical object, and $x$ a letter (i.e., a "totally indeterminate" mathematical object). In the assembly of letters and fundamental signs which constitutes the relation $R$, replace the letter $x$ wherever it occurs by the assembly $A$. One of the criteria for forming relations is that the assembly so obtained is again a relation, which is denoted $(\ast <!--\; \ast \; -->)$ by the notation
$(\mathrm{A}\mid <!--\; \mid \; -->x)\mathrm{R}$
and is called the relation obtained by substituting $A$ for $x$ in $R$, or by giving $x$ the value $A$ in $R$. The mathematical object A is said to satisfy the relation $R$ if the relation $(\mathrm{A}\mid <!--\; \mid \; -->x)\mathrm{R}$ is true. It goes without saying that if the letter $x$ does not appear at all in the assembly $R$, then the relation $(\mathrm{A}\mid <!--\; \mid \; -->x)\mathrm{R}$ is just $R$, and in this case to say that $A$ satisfies $R$ means that $R$ is true."

However, I do appreciate the textbook that is self-contained and appreciate the author devoted to mathematical reasoning so rigorously at the beginning of the chapter. I tried to find some textbook about mathematical logic but they are either too abstract or not thorough enough that seems to start from the most fundamental (i.e. from axiom and the most basic rule).
I have read relevant posts on the subject I am asking but can't decide the material right for me. I am wondering if there are any materials or textbooks that introduce mathematical logic rigorously and serve as a supplementary text for me to understand the first chapter of the book? If there really isn't any textbook that is not too abstract but rigorous enough, I am wondering if there are any other textbooks on abstract algebra that start from mathematical logic and build the whole system from the scratch?

Often when I'm working on a math problem, there comes a point when I've set everything up and what remains is to expand some expression, substitute something in, solve an equation, or otherwise enter the domain of algebra.
I find that I usually think about this step in the problem as a black box, into which I put a set of equations and out of which comes a set of solutions. This is particularly true if the algebra involved includes many steps.
This is good in a lot of ways and bad in some others, but putting that aside, I'm fascinated that this step seems to come up in all areas of math. Is it possible to separate math from algebra? Are there any branches of math where long chains of algebra are not found or uncommon?

So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel like I lack the intuition necessary to be able to come up with concepts on my own. I feel like I'm just missing some pivotal step in the journey to mathematical maturity.
Does anyone have any advice?
PS: If this means anything at all, as I was not a mathematics major, I did not take analysis or abstract algebra.

I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late university).
For example, I know that Stokes's theorem is a generalization of the divergence theorem, the fundamental theorem of calculus and Green's theorem, among I'm sure many other notions.
I've read that pure mathematics is concerned mostly with the concept of 'generalization' and I am wondering which theorems/ideas/concepts, like Stokes's theorem, are currently celebrated 'generalizations' by mathematicians.

For simple ones, like quadratic equations, I can usually find the minimum point and give a correct answer.
But take, for example:
$f(x)=\frac{3}{2x+1},x>0$
and
$g(x)=\frac{1}{x}+2,x>0$
I am so confused with the whole process of finding the ranges of functions, including those above as samples, that I can't even quite explain what or why.
Can someone please, in a very step-by-step process, detail exactly what steps you would take to get the ranges of the above functions? I tried substituting x-values (such as 0), and came up with $f(x)>3$, but that was mostly guesswork -- also, $f(x)>3$ is incorrect.
Also, is there an outline I can follow -- even for the thinking process, like check if A, check if B -- that would work every time?

Sometimes when solving very sparse equation systems
$Ax=b$
with conjugate gradient using computers, if A is a very sparse matrix, it can be difficult to utilize the hardware computational power maximally
Does there exist some way to rewrite
$Ax=b\text{ or }{A}^{T}Ax=A^{T}b$
to be able to utilize hardware better?
One idea I had is that often $A^2,A^3$ et.c. are increasingly non-sparse. Maybe it would be possible to take multiple steps at once..?
One motivation why this should be possible is that the Krylov subspaces which the Conjugate Gradient investigates are precisely the powers of the matrix. A second motivation why this is possible is of course Caley Hamilton theorem
$P(A)=0\Rightarrow <!--\; \Rightarrow \; -->A^{-<!--\; -\; -->1}=P_2(A)$
For some polynomial $P_2$ other than P.

Is there a genuinely respectable published expository account of what power series are used for, of which a substantial portion would be comprehensible to those first learning of power series in a first-year calculus course?
(I think it's a flaw in the way mathematics is conventionally taught that one first learns answers to questions like this only by taking a later course for which the one introducing the concept is a prerequisite, and one has no prior notice of which course that might be. And in the case of power series, there are many such courses, but still very few compared to the number of courses a student might later take, so the situation is worse.)

The tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised form of this for the direct sums of more than two vector spaces? Is there a relation between the space of all multilinear forms on the direct sum of $V_1$,$V_2$,$V_3$,...,$V_k$ with their tensor product.
Please explain without invoking other algebraic objects such as modules,rings etc and by using the concepts regarding vector spaces only (as the book assumes no such background either, it is unlikely that any reader of that book will benefit from such an exposition). Everywhere I searched, I found the explanation in terms of those concepts only and being unfamiliar to those I couldn't get them at all. Thanks in advance.