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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:

Y ax2 + bxn−1 + ... + kx + l

Polynomials are classified according to the degree of the highest power of x occurring.

Linear function

Quadratic function

Cubic function

Biquadratic function

ax + b.

ax2 + bx + c.

ax3 + bx2 + cx + d.

ax1 + bx3 + cx2 + dx + e.

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

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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

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x is also a function of y obtained by solving this equation for x, namely,

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The two functions x2 - 1 and ± √y + 1 are called inverse 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 Vx+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 a 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.

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3. Find the inverse of each of the following functions; classify each function and its inverse.

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4. Find the inverse functions defined by the following equations and classify them:

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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

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:

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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.

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2. Discuss the table of values, plot the graph, and determine the variation, including the average rate of change, of:

(a) 1x - 3.

(b) x2 - 4y

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:

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