Vocabulary Notes

1. So far - поки-що, до цього часу

2. We must alter our point of view a little - ми змушені дещо змінити свою точку зору

3. be ambiguous - буде двозначним (невизначеним)

4. an infinity of classes of entities - нескінченність класів ре¬альних речей

5. so far as mathematics is concerned - наскільки це стосується математики

Make a written translation of the text.

Text 4 functions

The main body of modern mathematics centers around the concepts of function and limit. An expression such as

x² + 2x — 3

has no definite numerical value until the value of x is assigned. We say that the value of this expression is a function of the value of x, and write

x² + 2x — 3 = f(x).

The number of primes less than n is a function π (n) of the integer n. When a value of n is given, 'the value π (n) is determined, even though no algebraic expression for computing it is known. The area of a triangle is a function of the lengths of its three sides. If a plane is subjected to a projective or a topological transformation, then the coordinates of a point after the transformation depend on, i. e. are functions of, the original coordinates of the point.

It may be that with each value of the variable X there is associated a definite value of another variable U. Then U is called a function of X. The way in which U is related to X is expressed, by a symbol such as

U = F(X) (read, "F of X").

Functions of a continuous variable are often defined by algebraic expressions. Examples are the functions ,

In the first and last of these expressions X may range over the

whole set of real numbers; while in the second, x may range over the set of real numbers with the exception of 0, the value 0 being excluded since 1/0 is not a number.

The number B (n) of prime factors of n is a function of n, where n ranges over the domain of all natural numbers. More generally , any sequence of numbers, a₁ a₂, a₃..., may be regarded as the set of values of a function,

u = F (n), where the domain of the independent variable n is the set of natural numbers.

If U= F (X) we usually reserve for X the name independent variable, while U is called the dependent variable, since its value depends on the value chosen for X.

It may happen that the same value of U is assigned to all values of X. We then have the special case where the value U of the function does not actually vary; that is, U is constant.

The concept of function is of the greatest importance, not only in pure mathematics but also in practical applications. Physical laws are nothing but statements concerning the way in which certain quantities depend on others. Thus the pitch of the note emitted by a plucked string depends on the length, weight, and tension of the string; the pressure of the atmosphere depends on the altitude. The energy of a bullet depends on its mass and velocity. The task of the physicist is to determine the exact or approximate nature of this functional dependence.

A mathematical function is simply a law governing the interdependence of variable quantities. The simplest types of mathematical functions of one variable are the polynomials, of the form

u = f (x) = a₀ + a₁x + a₂x² + ··· + a_n

with constant "coefficients" a₀, a, ...,a_n. Next come the rational functions, such as

which are quotients of polynomials, and the trigonometric functions, cos x, sin x, and tan x = The trigonometric functions are best defined by reference to the unit circle in the ξ, η: ξ 2 + η ²=1. If the point P (ξ, η) moves on the circumference of this circle, and if x is the directed angle through which the positive ξ -axis must be rotated in order to coincide with OP, then cos x and sin x are the

coordinates of P: cos x = ξ , sin x = η.

The character of a function is often most clearly shown by a simple geometrical graph. If x, y are coordinates in a plane with respect to a pair of perpendicular axes, then linear functions such as

u = ax +b

are represented by straight lines, quadratic functions such as

u=ax²+bx+c

by parabolas, the function

u=

by a hyperbola, etc.

By definition, the graph of any function u = f(x) consists of all the points in the plane whose coordinates x,u are in the relationship u = f (x).

An important method for introducing new functions is the following. Beginning with a known function, F(X),we may try to solve the equation U = F(X) for X, so that X will appear as a function of U:

X = G(U).

The function G(U) is then called an inverse function of F(X).The process leads to the unique result only if the function U = F(X) defines a biunique mapping of the domain of X onto that of U, i.e. if the inequality X₁X₂ implies the inequality F(X₁) F (X₂).Only then will there be a uniquely defined X correlated with each U.

Another example of biunique mapping is provided by the function

u = x³.

As x ranges over the set of all real numbers, u will likewise range over the set of real numbers, assuming each value once and only once. The uniquely defined inverse function is

X = .

In general, the inverse function exists and is uniquely defined if the function u = f(x) is monotone, i.e. steadily increasing as x increases. The graph of the inverse function x =g(u) is obtained merely by rotating the original graph through an angle of 180°, so that the positions of the x-axis and the u-axis are interchanged.