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!
Best Answer
Functions over Relations
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
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.