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Background

I've been interested lately in the idea of solving and inverting equations, and a question came to mind. Feel free to correct my notation, I could benefit from cleaner structuring. Going to stick primarily to the realm of real numbers here; just finished freshman year as a Math-Physics major, so I'm ill-equipped to deal with anything other than algebra and introductory calculus, but any and all answers are appreciated nonetheless.

Here's some context: in secondary education (high school), we typically learn about equation types of the simplest forms, starting off with equations such as - $$\begin{align} y &= ax + b\tag{1}\label{1}\\ y &= a^x\tag{2}\label{2}\\ y &= \sin(x)\tag{3}\label{3}\\ y &= ax^2 + bx + c\tag{4}\label{4} \end{align}$$ Where $a$,$b$,$c$ $\in{\mathbb{R}}$, $x$ is your input, and $y$ your output.

In all of these cases, one can set $y = 0$ and proceed to solve for $x$, or better yet, invert the functions altogether, setting $y$ in terms of $x$.

Here are those same examples, inverted - $$\begin{align} x &= \frac{y - b}{a}\tag{5}\label{5}\\ x &= log_{a}(y)\tag{6}\label{6}\\ x &= \sin^{-1}(y)\tag{7}\label{7}\\ x &= \frac{-b\pm{\sqrt{b^2-4a(c-y)}}}{2a}\tag{8}\label{8} \end{align}$$


Functions over Relations

The first aspect I'm struggling with understanding is why differentiate between a function and a relation? In other words - what good does that bring? In the case of \eqref{7} and \eqref{8}, their domains are restricted to allow them to continue being functions. Why do we do this? Why not just treat the two as relations instead of different inverse function branches?


Complexity of Inverting

In addition to the aforementioned, I was also curious about, with functions as simple as these, why does combining them seems to make inversion that much harder?

For instance, let's suppose you combined \eqref{2} and \eqref{4}, resulting in something of this form: $$y = a^x + bx^2 + cx + d$$ Right off the bat this looks incredibly hard to solve for $x$ when $y=0$, much less manipulating the equation to have it be in terms of $x$. Why is this the case, and what methods would you use to tackle problems like these?

(Note: I'm sure one could always define the inverse as the inverse of the function, just as how the square root is by definition the inverse function/relation of $ y = x^2 $, but I would prefer solutions in terms of elementary functions.)

Any intuitive and/or rigorous explanations help, thanks!

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  • $\begingroup$ Teaching x is input and Y is output is not teaching mathematics. It is teaching computer programming. $\endgroup$ Jun 6 '20 at 5:06
  • $\begingroup$ Oh, alrighty. What nomenclature should I use instead for pure mathematics purposes? $\endgroup$
    – Naganite
    Jun 6 '20 at 5:07
  • $\begingroup$ The value of f at x. $\endgroup$ Jun 6 '20 at 9:18
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Functions over Relations

why differentiate between a function and a relation?

Please note that mathematical objects are usually defined because their existence is a need to facilitate dealing with some math subjects. When you see that a math concept is often appeared in mathematical literature, you can conclude that it has a significant role and many applications inside and outside of (pure) mathematics.

The concept "relation" has been defined to show mathematical relationships between mathematical objects. A "function" is a special kind of a relation that for each input there is only one output; in fact, functions are well-behaved relations because we can control the outputs of a function by controlling its inputs, which is a very important fact in developing calculus subjects such as limit, differentiation, integration, and so on. Looking at various subjects inside and outside of (pure) mathematics, we can find that almost all relations are functions or can be written as a union of some functions.

For example, consider the circle $C:y=\pm \sqrt{1-x^2}$. This relation is not a function because for each input there are two outputs. However, we can write it as the union of the following functions:$$y=\begin{cases}f_1(x)=\sqrt{1-x^2} \\ f_2(x)=-\sqrt{1-x^2} \end{cases} \quad \Rightarrow \quad C= f_1 \cup f_2.$$Now, we can apply any facts about functions to the function pieces of a relation. Please note that almost all (applied) facts about functions are local properties, so we can use them to treat a relation as a function.

Please note that there is a general principle:

Generality of a concept is inversely related to the information we know about the concept.

This principle not only holds in mathematics but also in other branches of knowledge.

Functions are less general than relations, but we have much more information about functions than relations. Almost all facts in many various (applied) mathematics subjects are expressed in terms of functions, and most of them cannot be expressed in terms of relations, and even if they can, they become awkward; also as mentioned above, many relations can be written as a union of functions.


Complexity of Inverting

Why does combining them seems to make inversion that much harder?

I think the question is a special case of the following question:

Why are there many mathematics problems which most people can understand easily but have very difficult (or do not have) solutions?

The answer is, because we have a few known facts (Mathematicians call them "axioms") and we have to prove any results from them.

Let us change your mentioned example. The inverse of the functions $g(x)=x^5$ and $h(x)=-x$ can be easily found. So, why can we not find the inverse of the following function easily:$$f(x)=g(x)+h(x)=x^5-x$$(I only added two "elementary functions")?

Please note that there are some facts written in less than ten words but their proofs have hundred pages; there are also some facts which most people can understand but a few mathematicians can understand their proofs.

Mathematics is an axiomatic theory. It does not ask people to find easy proofs for its facts; it only wants them to prove results from only a few axioms.

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  • $\begingroup$ That was insightful, thanks! By any chance, do you know how the whole "unioning" of functions to get a relation works? A good example of relations that I've heard of being used often are elliptic curves: en.wikipedia.org/wiki/Elliptic_curve. How does one go about algebraically manipulating these equations to give you something useful (inversion being one such instance)? $\endgroup$
    – Naganite
    Jul 4 '20 at 17:12
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    $\begingroup$ @Naganite I edited my answer to add some explanation you asked in your comment. $\endgroup$
    – Later
    Jul 5 '20 at 6:46
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The Elementary functions are clearly defined. See e.g. MathWorld: Elementary function.
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Functions over Relations

When you write down equations like the ones listed in your question, all terms in your equation mean elementary functions: they represent the values of elementary functions. $\log(x)$ and $\sin(x)$ for example mean the values of the main branches of $Log$ and $Sin$ respectively.

If you invert your equations, you can represent the solutions of the equation in the picture of the single partial inverse functions or in the picture of the branches of the inverse relation.
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Complexity of Inverting

Each elementary function can be generated by applying finite numbers of $\exp$, $\log$ and/or unary, binary or multiary algebraic functions.

[Ritt 1925] and [Risch 1979] prove that the elementary functions that are invertible by elementary functions are the functions that are generated by applying finite numbers of $\exp$, $\log$ and/or unary algebraic functions. If the elementary function $H$ of your equation $H(x)=0$ is not invertible by an elementary function, you cannot solve (invert) the equation by rearranging it only by applying elementary partial inverses / elementary inverse operations of the elementary functions contained in the term H(x).

[Lin 1983] and [Chow 1999] prove that, if Schanuel's conjecture is true, an equation $P(x,e^x)=0$ with $P(X,Y)\in\overline{\mathbb{Q}}(X,Y)$ an irreducible polynomial involving $X$ and $Y$ cannot have a solution $x\neq 0$ that is an elementary number or an explicit elementary number respectively.

We can easily see, we get elementarily solvable equations if we combine equations of the right type, and we get non-elementarily solvable equations if we combine equations of the wrong type.
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[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759
[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90

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