EXAMPLE 2. A theorem from physics relating to a falling body states that if a heavy object be dropped, the distance it falls in t seconds is 16t2. This distance is a function of the variable t; for if t be given a definite positive value, the distance is determined. If t = 2 seconds, the distance fallen is 64 feet; if t = 3 seconds, the distance is 144 feet. Negative values of t are meaningless and hence not admissible. = EXAMPLE 3. The formula for the area of a circle is A Tr2. If the radius r be given a definite value the area is determined. Hence the area A is a function of r. The symbol represents the number 3.14159... which remains the same for any pair of corresponding values of r and A, and hence is called a constant in accordance with the DEFINITION. A constant is a symbol for a particular number. It has the same value throughout a discussion. EXAMPLE 4. A simple equation which occurs frequently in practice is y = mx, where x represents the independent variable, y the function, and m is a constant representing a fixed value as x and y vary. If x represents the number of pairs of shoes of a certain kind sold during a limited time by a dealer, and y the amount of the sales, then m represents the price per pair which remains fixed, for the time considered, while x and y vary. EXAMPLE 5. An equation in two variables establishes a functional relation between the variables. For if a value be given to either, the corresponding value or values of the other may be found by substituting the given value of one variable and solving for the other. Either variable may be regarded as a function of the other, and the form of the function may be found by solving for one variable in terms of the other. Thus, if the equation 4y - x2 = 0 be solved for y and then for x we have The given equation defines y as a function of x to be the function x2/4, and x as a function of y to be the function ±2√y. EXAMPLE 6. Some of the elements entering into the cost of a suit of clothes are the supply of cloth, the supply of labor, rent, style, etc. As these elements vary the cost of the suit will vary, so that the cost of the suit is a function of a number of variables. Considering one of the independent variables at a time, a part of the cost of the suit may be expressed as a function of this variable, e.g., the supply of cloth. The law of supply and demand from economics states that the price of an article increases or diminishes as the supply diminishes or in creases. If x represents the supply of cloth and y the price of the suit of clothes, then it is assumed in economics that the functional relation may be expressed by the equation where m is a constant which can be determined in any concrete case. In this course we shall confine ourselves to the study of functions of one variable. EXAMPLE 7. The temperature at a given place is a function of the time. For at a given time the temperature must have a definite value. But this function is so little understood that the Weather Bureau can only approximate the value for any future time, and that, indeed, only for times in the immediate future. The data of such departments of human knowledge as physics, astronomy, and engineering are so complete that many of the functions arising there can be identified and studied by mathematical methods. In other subjects, for example, chemistry and economics, the data have only recently been made sufficiently complete to warrant an increasing use of mathematics. But there still remains a countless number of functions which mankind has been unable to represent by a mathematical expression. EXERCISES In the following exercises give the reason for the statement that one variable is a function of another. 1. Mention three variables which are functions of the side of an equilateral triangle of varying size. 2. A train goes from one station to another at a variable rate. Mention two variables of which the rate is a function. 3. What are some of the variables of which the cost of erecting an office building is a function? 4. Mention some variables involved in heating water in a pan on a gas stove. Which are independent variables? Which are functions? 5. Find the functions of x defined by the following equations and tabulate three pairs of values. 5. Notation for a Function. It is convenient to represent a function of the variable x by the symbol f(x) which is read "function of x," "the function ƒ of x," or merely "f of x." f The various parts of the symbol are to be regarded as forming a single compound symbol, never as separate symbols meaning the product of two numbers ƒ and x. This symbol is used to denote either any function or a particular function such as f(x) = x2 + x − 1. Similar symbols convenient for distinguishing different functions are F(x), g(x), p(x), etc. An advantage of this notation is that the value of a function f(x) for any value of x, say x = a, may be suggestively represented by f(a). For example, if ƒ(−1) = (−1)2 + (−1) − 1 = − 1, ƒ(0) = − 1, ƒ(−x) = (−x)2 + (−x) − 1 = x2 − x − 1, etc. This notation also enables us to state certain theorems in a more compact form. EXERCISES 1. If f(x) = x3 – x2, find ƒ(1), ƒ( − 3), ƒ(0), ƒ(− x). 2. If F(x) = 1/x2, find F(2), F(− 1), F(a), F(− x). 3. If (x) mx + b, find (0), 4(1), P(x2), $(− b/m). = x3+x, show that ƒ(− 2) = – ƒ(2), that ƒ(− x) = − f(x). 4. If f(x) 5. If f(x) = x2 + x2 show that ƒ(− 2) = ƒ(2), that ƒ(− x) = f(x). 6. If f(x) x2 + x + 1, determine whether either f(x) = f(x) or = f(-x) 6. Determination of the Function which Expresses the Functional Relation between Two Variables. The functional relation between two variables is expressed in symbols whenever possible, for the sake of the greater simplicity which this gives to the generalization and of the greater ease in making deductions. If x is chosen to represent the independent variable, then an algebraic expression is sought to represent the function. x 0 EXAMPLE 1. If x represents any one of the numbers in the first column f(x) of the accompanying table, or the value of the independent variable, what function of x will represent the second column? We see by inspection that each number in the second set is the square of the corresponding number of the first set. 0 1 1 2 4 3 9 Check. Hence f(x) = x2 is the required function. When x = 0,ƒ(0) = 0; when x = 1, f(1) = 1, etc., for all the table. EXAMPLE 2. Find the area of an equilateral triangle as a function of the side. Let x represent the side of the equilateral triangle ABC with the altitude CD. C √3 x = x2, which is the required 4 function of x. EXAMPLE 3. A man leaves a village 10 miles directly east of a city and walks east at the rate of 3 miles an hour. Express the distance he is from the city at any time, t, after he starts as a function of t. Where will he be at the end of 3 hours, and when will he be 25 miles from the city? t S rate Since distance equals the rate multiplied by the time, the distance the man is from the village at any time t after he starts is 3t; and the distance he is from the city is = 3t+ 10, since the village and city are 10 miles apart and he goes away from both. Hence the required function is 3t+ 10. When t The data and results are recorded in the accompanying table. EXAMPLE 4. A ball rolls down an inclined plane. The distance s it rolls in the tth second is recorded in the table. Express the distance the ball rolls in any second as a function of t. What distance will it roll in the 5th second? t 8 1 4 2 12 3 20 The values of both the independent variable and the function form arithmetic progressions. The formula for the nth term of an arithmetic progression is l = a + (n − 1) d. The common difference d of the values of the function is 8, the first term a is 4, the number of terms n is t, and l = 8 1 Check. If the values 1, 2, 3 are substituted for t in 8t 4 the values 4, 12, 20 are obtained, which are the values of s given in the table. EXAMPLE 5. If $100 are placed in a bank at 5% interest, compounded annually, what is the amount at the beginning of tth year? The amount at the beginning of the first year is of course $100. The interest for the first year is $5 and the amount at the beginning of the second year is $105. t A 100 2 105 3 110.25 The interest for the second year is $5.25 and the gression, for the ratio of the second value to the first is equal to the ratio of the third value to the second, i.e., The nth term of a geometric progression is given by the formula In this case = A, n = t, a = 100 and r = 1.05. Substituting these values in the above formula, we have for the required function Check. Substituting the values 1, 2, 3, for t in this function we get the values 100 105 and 110.25 respectively for A, which results agree with the table. EXERCISES 1. In each of the following tables let x represent any one of the numbers in the first column (value of the independent variable), and find a function of x which will represent the corresponding number in the second column. Add two additional pairs of values of x and the function to each table. |
