Functions and Graphs

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

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486317137
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

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.

CHAPTER 1

Examples

1

If the definition is followed literally, then in order to construct a graph of some function, it is necessary to find all pairs of corresponding values of argument and function and to construct all points with these co-ordinates. In the majority of cases it is practically impossible to do this, since there are infinitely many such points. Therefore, usually a few points belonging to the graph are joined by a smooth curve.

In this way, let us try to construct the graph of the function

Table 1

Let us choose some values of the argument, find the corresponding values of the function, and write them down in a table (see Table 1). We construct the points with the computed coordinates and join them by a dotted line, for the time being (Fig. 1).

Fig. 1

Let us now verify whether we have drawn the curve correctly between the points found to lie on the graph. For this purpose let us take some intermediate value of the argument, say,

, and compute the corresponding value of the function

. The point obtained,

, falls nicely on our curve (Fig. 2), so that we have drawn it quite accurately.

Fig. 2

Now we try

. Then

, and the corresponding point lies above the curve we have drawn (Fig. 2). This means that between x = 0 and x = 1 the graph does not go as we thought. Let us take two more values,

and

, in this doubtful section. After connecting all these points, we get the more accurate curve represented in Fig. 3. The points

and

, taken as a check, fit the curve nicely.

Fig. 3

2

In order to construct the left half of the graph, it is necessary to fill in one more table for negative values of the argument. This is easy to do. For example,

This means that together with the point

, the graph also contains

, the point symmetric to the first with respect to the y-axis.

In general, if the point (a, b) lies on the right half of our graph, then its left half will contain the point (−a, b) symmetric to (a, b) with respect to the y-axis (Fig. 4). Therefore, in order to obtain the left part of the graph of function (1) corresponding to negative values of x, it is necessary to reflect the right half of this graph in the y-axis.

Fig. 4

Figure 5 shows the over-all form of the graph.

If we had been hasty and had used our original sketch for the construction of the part of the graph corresponding to negative x (Figs. 1 and 2), then it would have had a “kink” (corner) at x = 0. There is no such kink in the accurate graph; instead there is a smooth “dome.”

If the values of some function corresponding to any two values of the argument equa...

Table of contents

• Cover
• Title
• Copyrights
• Foreword
• Contents
• Introduction
• Chapter 1: Examples
• Chapter 2: The Linear Function
• Chapter 3: The Function Y = X
• Chapter 4: The Quadratic Trinomial
• Chapter 5: The Linear Fractional Function
• Chapter 6: Power Functions
• Chapter 7: Rational Functions
• Problems for Independent Solution
• Answers and Hints to Problems and Exercises Marked by the Sign ⊕