For a temperature of 400°, or greater, an increase of 200° in the temperature causes an expansion of approximately 0.1 of a millimeter in the length of the rod. The generalization may also be expressed in symbols. If every increase of 200° causes an expansion of 0.1 of a millimeter, then an increase of 1° would cause an expansion of 0.1/200, or 0.0005; hence for an increase of temperature of t degrees the expansion e would be e = 0.0005t. This equation expresses the generalization in more compact form than the sentence above. In this illustration we have considered only a part of the table, a part for which the generalization is very simple. The determination of the mathematical expression of the generalization from a table of values will be one of the objects of this course. The generalization from a mathematical point of view is considered in the following section. 4. Variable. Function. The following table gives the lengths of an iron bar suspended from one end when carrying different loads. Any one of the numbers in each of these sets of numbers is conveniently represented by a single symbol, as x, y, t, etc. Thus x may be taken to represent one of the numbers 0, 500, 1000, 1500, 2000 and any other number that might be included in the half of the table giving loads, while y may represent the corresponding number giving lengths in the second part of the table. The symbols x and y are called variables, in accordance with the DEFINITION. A variable is a symbol for any one of a set of numbers. In the experiment giving rise to the above table, the load was changed arbitrarily by the experimenter and the length of the bar for the chosen load measured. In consequence of this order of measurement the variable representing the load is called the independent variable and the one representing the length of the bar the dependent variable. It is customary to denote the independent variable by x and the dependent variable by y. If the experiment were repeated, under the same conditions, it would be found that for a specified load the length of the bar would be the same, that is, there is a law connecting the load and the length of the bar. This relation between the variables is expressed by saying that the length, y, is a function of the load, x, in accordance with the DEFINITION. A function is a variable so related to another variable (called the independent variable) that for every admissible value of the independent variable, one or more values of the function are determined. The function is also called the dependent variable. The idea of a function arises wherever there is a relation between magnitudes which are changing, and it underlies all magnitude relations which mankind has discovered. EXAMPLE 1. The algebraic expression 2x + 3 is a variable whose value is determined whenever a definite value is assigned to x. If x be given the value 1, then 2x + 3 has the value of 5, and if x has the value 2, 2x + 3 has the value 7. Hence 2x + 3 is a function of x. 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 16ť2. 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. 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 π 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 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 tabufate three pairs of values. = (a) y − x2 3x 2 0. (b) x3 – 3y2 + 2y + 4 = 0. 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 f of x," or merely "f of x." 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 f 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 f(x) = x2 + x − 1, f(a) = a2 + a − 1, ƒ(2) 22 +215, ƒ(−1) = (−1)2 + (−1) − 1 ƒ(−x) = (−x)2 + (−x) − 1 = − 1, ƒ(0) = − 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), 4(x2), $(− b/m). = 4. If f(x) = x3 +x, show that f(− 2) = − ƒ(2), that ƒ(− x) = − f(x). f(x) or 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 |