tional relation (Sections 1-4). If this relation can be expressed in mathematical symbols, then mathematics becomes, to a considerable extent, the language of that science. An important problem, therefore, is the determination of an expression in mathematical symbols for a function which is given otherwise (Section 6; see also 5 and 6 below). Function - table of values-graph. These are merely different manifestations of the same concept. A functional relation expresses a law connecting each pair of numbers of two sets; the table gives particular pairs of these numbers; and the graph affords a geometric representation of them. The fundamental problems arising in connection with this concept are the determinations of any two of these manifestations from the third. They are six in number. 1. Given a function, to build the table of values. This is an easy process for simple algebraic functions. 2. Given a table of values, to plot the graph. This is also easily done if the table is sufficiently extensive. 3. Given a function, to draw the graph. This is done by means of 1 and 2 (Section 9). 4. Given a graph, to construct a table of values. Pairs of values may be read, approximately, directly from the graph. 5 and 6. Given a table of values, to find the function. This is precisely the problem which confronts the scientist in seeking an unknown law, and it is by far the most difficult of these problems. Methods of solution will be considered in the following chapters (see Sections 25, 44, and 90). In Sections 10-13 we have considered properties of a function, its table, and its graph, which are important in the study of any function. The correspondence between these properties of a function and its graph may be exhibited compactly as follows: PROPERTY OF GRAPH Length of an ordinate. Graph symmetrical respect y-axis. PROPERTY OF FUNCTION A value of the function. ƒ(− x) = − f(x). Excluded values of x. Function imaginary or infinite. Interpretation of a graph. The properties of the graph may be determined by inspection, if only one has in mind what to look for, and the corresponding properties of the function may then be stated. It is true that the properties of the graph are proved by first establishing the properties of the function. But after the graph is drawn, this interpretation of the graph affords a comprehensive point of view of many properties of the function and furnishes a simple means of stating any particular property. As we come to study any one class of functions defined in Section 14, we shall take up the properties listed above, and in addition the characteristic properties which distinguish that class of functions from others. A typical graph for each class of functions should be fixed in mind, as it enables one to tie together and recall quickly the characteristics of a function as soon as it is classified. This is of great importance in analyzing problems. From time to time we shall add other properties of graphs and functions to the list above. We shall also take up the relations between certain pairs of functions and their graphs, which enable us to obtain the graph of one function from that of the other. Thus the graph of f(x) + k may be obtained from that of f(x) (see Exercise 3, page 19). These two sets of relations constitute the framework of the entire course. We turn next to the detailed study of particular classes of functions, beginning with the linear function, which occurs frequently in the applications of mathematics. MISCELLANEOUS EXERCISES 1. State as many properties as possible of the functions whose graphs are given below. 2. Discuss the table of values, plot the graph, and determine the variation, including the average rate of change, of: (a) x 3. (b) x2 - 4y - 3 = 0. (c) x3 16x + 3y = 0. 3. Discuss the table of values, plot the graph, and determine the variation, omitting the average rate of change, of: (g) y2 = x3. (h) x2y4y-10. (i) x2y 4y x = 0. 16 0. (1) xy2+ x - 4y = 0. (j) y2(4 − x) = x2(4 + x). (k) x3 + 8x บ 4. For a given abscissa, the ordinate of a point on the graph of y = f(x) + g(x) is the sum of the ordinates of the points with the same abscissa on the graphs of f(x) and g(x). Hence the graph of y may be obtained by drawing the graphs of f(x) and g(x) and then adding the ordinates of points with the same abscissa. Using this method, which is called the addition of ordinates, construct the graphs of the following functions. R -3 P 2 R M M FIG. 26. P (a) y = x2 + 1x. Solution. Draw the graphs of x2 and of x. If MP and MQ are the ordinates of points on these graphs with the same abscissa OM, then MR = MP + MQ = MP + PR is the ordinate of a point on the graph of the given function. Several points on the graph may be obtained in this way, and a smooth curve drawn through them. Notice that if OM is negative, so also is MQ, so that MR<MP. (b) x2 + x. (e) x3/3 + x/3. (c) x2/3 + 2x. (f) 1/x + x/2. (d) x2/2+x. (g) x2 - 2x. 5. Express the area of a rectangle as a function of one of its sides, assuming that the perimeter is 8 feet. Plot the graph and discuss the variation of the function. 6. A house stands 50 feet from the street. A man is walking along the street. Express his distance from the house as a function of his distance from the entrance to the grounds, which is directly in front of the house. Discuss and plot the graph of the function. 7. A man 6 feet tall walks away from a lamp-post 12 feet high. Express the length of his shadow as a function of his distance from the post, and plot the graph of the function. 8. In Exercise 7, express the distance from the post to the shadow of the man's head as a function of the man's distance from the post. Plot the graph, and show that the average rate of change of the function is constant. 9. A point lies at a distance r from the origin. Find the equation expressing the functional relation between the coördinates x and y of the point. 10. A man walks for 2 hours at the rate of 3 miles an hour, stops an hour and a half for lunch, and walks back at the rate of 2 miles an hour. Construct a graph showing his distance from the starting point at any time. NOTE. A solution of two simultaneous equations in x and y consists of a pair of values of these variables which satisfy both equations. If the numbers are real, the point whose coördinates are such a pair of values is on both graphs. Therefore a real solution of two simultaneous equations in x and y is represented by a point of intersection of the graphs. To find the coördinates of the points of intersection of two graphs, solve their equations simultaneously. 11. Plot the graphs of the following equations and find the coördinates of their points of intersection. 12. Plot the graphs of the following equations and from them read approximate values of the solutions of the equations. 13. Two engines in a freight yard are running on parallel tracks. Their distances from a signal tower (in hundred yards) at the time t (in minutes) are given respectively by the equations |
