Brief Overview of Functions

After we have a set, the next thing we need is a function. For most of mathematics, this is all that is needed. The rest is just studying the relation between the sets and the functions. So what is a function?

A function is a rule and a set. The rule associates an element of the set to exactly one element of another set, which could be the same set.
In particular, I’m going to consider sets of numbers although the sets could be groups, rings, or any number of other kinds of sets.
A function is actually a pair, a rule and a set. The set is often not mentioned explicitly but it is a crucial part of the definition.
Suppose that the set is the set of numbers between 0 and 10 inclusive, that is [1,10]. The rule is to associate a number in the set with its square. I take a number out of [0,10] and square it. The rule associates 5 with 25. The set of numbers that the rule is applied to is called the domain of the function.
I can use algebra to express the rule. Associate a number x in the domain with the number x². As is usual we give the rule a letter name, say f. We denote the number that the rule of f, associates with the number, x, by f(x). So we can write
f(x) = x².
(f(x) is not a function, it is a number. The function is f where f stands for the rule and its domain. I take a number from the domain and associate it with its square.)
[0,10] could represent the points along a 10 cm. rod. The function, f, could associate a point on the rod to the temperature of the rod at that point in degrees C. Suppose that the temperature at x is f(x) = x² so the temperature at x = 5 is 25°C.
If I have a rod 20 cm. long, the rule that gives the temperature at x could be the same as the rule of f, the temperature at x is x². But the domain of this function is [0,20] and so it is a different function than f and must be given a new name, say g.
g(x) = x² because f and g have the same rules but they are different functions because they have different domains.
Next we examine the behavior of functions, in particular how do they act on the operations defined on the set. Since I am looking at a set of numbers, whose operations are +, -, • and ÷, I look at

f(x + y), f(x - y), f(x•y) and f(x/y).
It is quite possible that they are nothing special but, on the other hand, they could be special.
Maybe f(x + y) = f(x) + f(y). Maybe f(x•y) = f(x) f(y). Maybe not.
In trigonometry there are formulae for sin(x + y) and cos(x + y). These formulae are just examples of how the functions sin and cos, whose domain I take to be all numbers, behave with the operation of addition.
There are other properties that a set may have and functions relate to. For example there can be a distance between two numbers.
Here I’m thinking of the numbers on a number line and the distance between two numbers, x and y, is
|x-y|
This is the distance you would find with a tape measure.
If x and y are close together, are f(x) and f(y) close together? How does the distance between f(x) and f(y) relate to the distance between x and y? This gives rise the concept of continuity. If we examine this relationship a little closer, we look at the ratio
f(x) - f(y) / (x - y).
This gives rise to the concept of the derivative which essentially describes how fast f(x) changes as x changes. By introducing the idea of graphs, we get a picture of how f(x) changes as x changes.
My point here is not the details of the analysis of the relationship between x and f(x) but the kind of questions that are asked.
This type of overview is something that most courses in mathematics leave out. One of the most egregious examples is the introduction of the epsilon-delta process. The student is asked to show that that sin x is continuous using epsilons and deltas when it is obvious that it is continuous. Until the time of Weierstrass mathematicians assumed that all functions were piecewise continuous and they did just fine, particularly with the kind of mathematics the undergraduate encounters. Undergraduate mathematics barely gets out of the 18th century.
When I was in graduate school, at the beginning of each semester an advanced student would tell me what was going to happen in the courses I had signed up for. Then I wasn’t driving 90 mph down an unknown highway at night with my dims on.
I would suggest that a course in mathematics start by telling the students what the point of the course is, what the instructor wants to accomplish and why.