responding values of the independent variable and the function, and hence these values satisfy the given equation. The average rate for any particular interval may be obtained from this result by substitution. For example: For the first given interval, from 1 to 3, x starts with the value x = 1, and increases by Ax = 2. Substituting these values in (1), we get As these results agree with those obtained above, we have a desirable check on the correctness of the entire procedure. The graphical representation of the average rate of change is important. The corresponding pairs of values (x, y) and (x + Ax, y + ▲y) are represented by two points P and Q on the graph, so that x = OM, y = MP, x + Äx = ON, y + ▲y = NQ. EXERCISES 1. Find the average rate of change of each of the functions below if x changes from 0 to 1; from 1 to 3; from 1 to 4. What can be said of these average rates? Find the average rate for any interval, draw the graph, and interpret the average rate graphically (a) 2x 3. (d) x + 2. (b) 2x. (e) - 3x + 3. (c) (f) - 2x + 5. -x +3. 2. Find the average rate of change of each of the following functions for the intervals from 1 to 3; from 1 to -1; from 0 to 3; for any interval. Check the results. Plot the graph and interpret the average rates. 3. For each of the following functions, discuss the table of values, plot the graph, and determine the variation, including the average rate of change 4. Discuss the function represented by the following temperature graph. Find the greatest and the least average rate for the three-hour intervals indicated on the time axis. 40 30° 20° 10° 0° 12 M. 3 A.M. 6 A.M. 9 A.M. 12 Noon 3 P.M. 6 P.M. 9 P.M. FIG. 24. 12 M. 5. The daily variation of temperature for a person in normal condition is given in the following table. Plot the graph and discuss the function. For what hour of the day is the average rate the greatest? The least? The temperature from 2 A.M. to 6 A.M. is stationary. 14. Classification of Functions. In discussing functions it is convenient to separate them into classes according to the properties they possess or the character of the operations that are involved in calculating them. The following scheme indicates the important divisions and subdivisions of the functions which we shall study in this course. * An algebraic function is a function whose value may be computed when that of x is given by the application, a finite number of times, of the operations of algebra, namely, addition, subtraction, multiplication, division, involution. and evolution. The following are examples of algebraic functions: 2x + 3, ax2 + bx + c, √x2 − 9, 2x + 5 x 3 In the work of this course only real numbers will be used. Hence, for us, such an algebraic function as V16 x2 is defined only for values of x such that -4x + 4 (read "x is greater *This definition is sufficiently general for the purposes of this course. The definition used in higher mathematics, which is more inclusive is as follows: Given a polynomial in y, whose coefficients ao, a1, a2, an-1, an are polynomials in x, tnen any solution of the equation f(y) = 0 for y in terms of x is called an algebraic function. than or equal to 4, and less than or equal to +4"), since for all other values, as x = 5, the value of the function is imaginary. An integral rational function (integral function or polynomial) is a function whose value, for a given value of x, may be found by the operations of addition, subtraction, and multiplication applied a finite number of times. The following is the general form: Polynomials are classified according to the degree of the highest power of x occurring. A rational function is a function whose value, for a given value of x, may be found by the four rational operations of addition, subtraction, multiplication, and division, applied a finite number of times. If division involving x in the divisor occurs in the computation, the function may be expressed as the quotient of two polynomials, and it is then called a fractional function. The general form is y = ax" + ax-1+ + An-1x + An box + b1xm-1+ ... + bm-1x + bm An irrational function is a function which involves tne extraction of roots in addition to the four rational operations. In contrast with algebraic functions, all other functions are called transcendental. This term is merely a synonym for non-algebraic. Inverse functions. If y be given as a function of x by the relation x is also a function of y obtained by solving this equation for x, namely, X = ± √y +1. Vy+1 are called inverse The two functions x2 1 and ± functions. The independent variable in the first is, as usual, x, but in the second it is y. In order to write the inverse function with x as the independent variable we must replace y by x. Hence the inverse of x2 - 1 is ±√x + 1. DEFINITION. If y be set equal to f(x), the equation solved for x in terms of y, and y replaced by x in the final result, then the function of x so obtained is called the inverse of f(x). The same result is obtained by interchanging x and y in the equation y = f(x), and then solving for y in terms of x. 3. Find the inverse of each of the following functions; classify each function and its inverse. 4. Find the inverse functions defined by the following equations and classify them: 15. Summary. Suppose that the data of a law of a science can be recorded in a table of values of two variables, or, by some mechanical device, in the form of a curve. The generalization which expresses the law connecting the two variables is a func |
