Text b. Functions

We now turn to the discussion of the fundamental notion of a function or mapping. It will be seen that a function is a special kind of a set, although there are other visualizations which are often suggestive.

To the mathematician of a century ago the word “function” ordinarily means a definite formula, such as f(x) = x² + 3x + 5, which associates to each real numberxanother real number f(x). The fact that certain formulas, such as

g(x) = √x – 5, do not give rise to real numbers for all real values of x was, of course, well-know but was not regarded as sufficient grounds to require an extension of the notion of function. Probably one could arouse controversy among those mathematicians as to whether the absolute valueh (x) = │x│of a real number is an “honest function” or not. For, after all, the definition of│x│is given “in pieces” by

│x│ = x, if x ≥ 0, │x│ = - x, if x < 0.

As mathematics developed, it became increasingly clear that the requirement that a function be a formula was unduly restrictive and that a more general definition would be useful. It also became evident that it is important to make a clear distinction between the function itself and the values of the function.

Our first revised definition of a function would be:

A function ffrom a setAto a setBis a rule of correspondence that assigns to each x in a certain subsetDofA, a uniquely determined elementf (x) ofB.

Certainly, the explicit formulas of the type mentioned above are included in this definition. The proposed definition allows the possibility that the function might not be defined for certain elements of Aand also allows the consideration of functions for which the setAandBare not necessarily real numbers.

However suggestive the proposed definitions may be, it has a significant defect: it is not clear. There remains the difficulty of interpreting the phrase “rule of correspondence”. The most satisfactory solution seems to define “a function” entirely in terms of sets and the notions introduced above.

The key idea is to think of the graph of the function, that is, a collection of the ordered pairs.

Definition.LetAandBbe sets. A function fromAtoBis a setfof ordered pairs inAxBwith the property that if(a, b) and(a', b')are elements off, thenb = b'. The set of all elements ofAthat can occur as first members of elements infis called the domain offand will be denotedD (f).The set of all elements ofBthat can occur as second members of elementsfis called the range off(or the set of values off) and will be denoted byR (f). In caseD (f)= A, we often say thatfmapsAintoB(or is a mapping ofAintoB) and writef :A → B.

If (a, b) is an element of a function f, then it is customary to write b = f (a) or f : a → b instead of (a, b) Є f. We often refer to the element b as the value of f at the point a, or the image under f of the point a.

Ex. 20. Say these sentences in English.

Основные понятия функции.

Если каждому значению х из некоторой области D поставлено в соответствие значение переменной у, то говорят, что в области D задана функция у аргумента х:

y = f(x)

Это типичное обозначение функции. Область D называется областью определения функции, а совокупность значений переменной у - областью ее изменения. Уравнение у = f(x) можно интерпретировать графически как уравнение кривой в х, у - плоскости. Говорят, что функция f задает отображение множества X на множество Y, если для любого y є Y существует такое х є Х, что f(x) = у. Это отображение является взаимно однозначным, если из равенства f(x) = f(z) следует, что

x = z.

Функции можно также задавать с помощью таблиц.

Примером такого задания является нижеприведенная таблица.

Таблица 1

x	-3	-2	-1	0	1	2	3
f (x)	5	2	0	-1	3	4	5

Кроме того, зависимость между переменными х и у можно задавать графически. Каждая пара чисел в вышеприведенной таблице задает точку в плоскости х 0 у. Если нанести эти точки и соединить их плавной кривой, то мы получим график функции у = f (х).