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    Functions and Graphs
    Functions and Graphs
    Functions and Graphs
    Ebook220 pages1 hour

    Functions and Graphs

    By I.M. Gelfand, E. G. Glagoleva and E. E. Shnol

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    About this ebook

    The second in a series of systematic studies by a celebrated mathematician I. M. Gelfand and colleagues, this volume presents students with a well-illustrated sequence of problems and exercises designed to illuminate the properties of functions and graphs. Since readers do not have the benefit of a blackboard on which a teacher constructs a graph, the authors abandoned the customary use of diagrams in which only the final form of the graph appears; instead, the book's margins feature step-by-step diagrams for the complete construction of each graph. The first part of the book employs simple functions to analyze the fundamental methods of constructing graphs. The second half deals with more complicated and refined questions concerning linear functions, quadratic trinomials, linear fractional functions, power functions, and rational functions.
    LanguageEnglish
    PublisherDover Publications
    Release dateApr 9, 2013
    ISBN9780486317137
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      Book preview

      Functions and Graphs - I.M. Gelfand

      ⊕

      Introduction

      Fig. 1

      In Fig. 1 the reader can see two curves traced by a seismograph, an instrument which registers fluctuations of the earth’s crust. The upper curve was obtained while the earth’s crust was undisturbed, the lower shows the signals of an earthquake.

      In Fig. 2 there are two cardiograms. The upper shows normal heartbeat. The lower is taken from a heart patient.

      Fig. 2

      Figure 3 shows the so-called characteristic of a semiconducting element, that is, the curve displaying the relationship between current intensity and voltage.

      Fig. 3

      In analyzing a seismogram, the seismologist finds out when and where an earthquake occurred and determines the intensity and character of the shocks. The physician examining a patient can, with the help of a cardiogram, judge the disturbances in heart activity; a study of the cardiogram helps to diagnose a disease correctly. The characteristic curve of a semiconducting element enables the radio-electrical engineer to choose the most favorable condition for his work.

      All of these people study certain functions by the graphs of these functions.

      What, then, is a function, and what is the graph of a function ?

      Before giving a precise definition of a function, let us talk a little about this concept. Descriptively speaking, a function is given when to each value of some quantity, which mathematicians call the argument and usually denote by the letter x, there corresponds the value of another quantity y, called the function.

      Thus, for example, the magnitude of the displacement of the earth’s surface during an earthquake has a definite value at each instant of time; that is, the amount of displacement is a function of time. The current intensity in a semiconducting element is a function of voltage, since to each value of the voltage there corresponds a definite value of current intensity.

      Many such examples can be mentioned: the volume of a sphere is a function of its radius, the altitude to which a stone rises when thrown vertically upward is a function of its initial speed, and so on.

      Let us now pass to the precise definitions. To say that the quantity y is a function of the quantity one first of all specifies which values x: can take. These allowed values of the argument x are called admissible values, and the set of all admissible values of the quantity or variable x is called the domain of definition of the function y.

      For instance, if we say that the volume V of the sphere is a function of its radius R, will be all numbers greater than zero, since the value R of the radius of the sphere can be only a positive number.

      Whenever a function is given, it is necessary to specify its domain of definition.

      Definition I

      We say that y is a function of the variable x, if: (1) it is specified which values of x are admissible, i.e., the domain of definition of the function is given, and (2) to each admissible value of x there corresponds exactly one value of the variable y.

      Instead of writing "the variable y is a function of the variable x", one writes

      (Read: "y is equal to f of x.")

      The notation f(a) denotes the numerical value of the function f(x) when x is equal to a. For example, if

      then

      The rule by which for each value of x the corresponding value of y is found can be given in different ways, and no restrictions are imposed on the form in which this rule is expressed. If the reader is told that y is a function of x, then he has only to verify that: (1) the domain of definition is given, that is, the values that x can assume are specified, and (2) a rule is given whereby to each admissible value of x there can be associated a unique value of y.

      What kind of rule can this be?

      Let us mention some examples.

      1. Suppose we are told that x may be any real number and y can be found by the formula

      The function y = x² is thus given by a formula.

      The rule may also be verbal.

      The function y is given in the following manner: If x is a positive number, then y is equal to 1; if x is a negative number, then y is equal to −1; if x is equal to zero, then y is equal to zero.

      Let us mention yet another example of a function given by a verbal rule.

      3. Every number x can be written in the form x = y + α where α is a nonnegative number less than one, and y is an integer. It is clear that to each number x there corresponds a unique number y; that is, y is a function of x. The domain of definition of this function is the entire real axis. This function is called "the integral part of x" and is denoted thus:

      For example,

      [3.53] = 3, [4] = 4, [0.3] = 0, [−0.3] = −1.

      We shall use this function later in our exercises.

      4. Let us consider the function y = f(x), defined by the formula

      What can reasonably be considered as its domain of definition?

      If a function is given by a formula, then usually its so-called natural domain of definition is considered, that is, the set of all numbers for which it is possible to carry out the operations specified by the formula. This means that the domain of definition of our function does not contain the number 5 (since at x = 5 the denominator of the fraction vanishes) and values of x less than −3 (since for x < −3 the expression under the root sign is negative). Thus, the natural domain of definition of the function

      consists of all numbers satisfying the relations:

      A function can be represented geometrically with the help of a graph. In order to construct a graph of some function, let us consider some admissible value of x and the corresponding value of y. For example, suppose the value of x is the number a, and the corresponding value of y is the number b = f(a). We represent this pair of numbers a and b by the point with the coordinates (a, b) in the plane. Let us construct such points for all admissible values of x. The collection of points obtained in this way is the graph of the function.

      Definition II

      The graph of a function is the set of points whose abscissas are admissible values of the argument x and whose ordinates are the corresponding values of the function y.

      For example, Fig. 4 depicts the graph of the function

      Fig. 4

      It consists of an infinite set of horizontal line segments. The arrows indicate that the right end points of these segments do not belong to the graph (whereas the left end points belong to it and therefore are marked by a thick point).

      A graph can serve as the rule by which the function is defined. For example,

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