Chapter II
Functions of Real Variables

20. The idea of a function. Suppose that $x$ and $y$ are two continuous real variables, which we may suppose to be represented geometrically by distances $A_{0}P=x$, $B_{0}Q=y$ measured from fixed points $A_0$, $B_0$ along two straight lines $\Lambda$, $M$. And let us suppose that the positions of the points $P$ and $Q$ are not independent, but connected by a relation which we can imagine to be expressed as a relation between $x$ and $y$: so that, when $P$ and $x$ are known, $Q$ and $y$ are also known. We might, for example, suppose that $y=x$, or $y=2x$, or $\frac{1}{2}x$, or $x^2+1$. In all of these cases the value of $x$ determines that of $y$. Or again, we might suppose that the relation between $x$ and $y$ is given, not by means of an explicit formula for $y$ in terms of $x$, but by means of a geometrical construction which enables us to determine $Q$ when $P$ is known.

In these circumstances $y$ is said to be a function of $x$. This notion of functional dependence of one variable upon another is perhaps the most important in the whole range of higher mathematics. In order to enable the reader to be certain that he understands it clearly, we shall, in this chapter, illustrate it by means of a large number of examples.

But before we proceed to do this, we must point out that the simple examples of functions mentioned above possess three characteristics which are by no means involved in the general idea of a function, viz.:

(1) $y$ is determined for every value of $x$;

(2) to each value of $x$ for which $y$ is given corresponds one and only one value of $y$;

(3) the relation between $x$ and $y$ is expressed by means of an analytical formula, from which the value of $y$ corresponding to a given value of $x$ can be calculated by direct substitution of the latter.

It is indeed the case that these particular characteristics are possessed by many of the most important functions. But the consideration of the following examples will make it clear that they are by no means essential to a function. All that is essential is that there should be some relation between $x$ and $y$ such that to some values of $x$ at any rate correspond values of $y$.

Examples X. 1. Let $y=x$ or $2x$ or $\frac{1}{2}x$ or $x^2+1$. Nothing further need be said at present about cases such as these.

2. Let $y=0$ whatever be the value of $x$. Then $y$ is a function of $x$, for we can give $x$ any value, and the corresponding value of $y$ (viz. $0$) is known. In this case the functional relation makes the same value of $y$ correspond to all values of $x$. The same would be true were $y$ equal to $1$ or $-\frac{1}{2}$ or $\sqrt{2}$ instead of $0$. Such a function of $x$ is called a constant.

3. Let $y^2=x$. Then if $x$ is positive this equation defines two values of $y$ corresponding to each value of $x$, viz. $\pm \sqrt{x}$. If $x=0$, $y=0$. Hence to the particular value $0$ of $x$ corresponds one and only one value of $y$. But if $x$ is negative there is no value of $y$ which satisfies the equation. That is to say, the function $y$ is not defined for negative values of $x$. This function therefore possesses the characteristic (3), but neither (1) nor (2).

4. Consider a volume of gas maintained at a constant temperature and contained in a cylinder closed by a sliding piston.¹⁹

Let $A$ be the area of the cross section of the piston and $W$ its weight. The gas, held in a state of compression by the piston, exerts a certain pressure $p_0$ per unit of area on the piston, which balances the weight $W$, so that

Let $v_0$ be the volume of the gas when the system is thus in equilibrium. If additional weight is placed upon the piston the latter is forced downwards. The volume ($v$) of the gas diminishes; the pressure ($p$) which it exerts upon unit area of the piston increases. Boyle’s experimental law asserts that the product of $p$ and $v$ is very nearly constant, a correspondence which, if exact, would be represented by an equation of the type

where $a$ is a number which can be determined approximately by experiment.

Boyle’s law, however, only gives a reasonable approximation to the facts provided the gas is not compressed too much. When $v$ is decreased and $p$ increased beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation (i). It is known that a much better approximation to the true relation can then be found by means of what is known as ‘van der Waals’ law’, expressed by the equation

where $\alpha$, $\beta$, $\gamma$ are numbers which can also be determined approximately by experiment.

Of course the two equations, even taken together, do not give anything like a complete account of the relation between $p$ and $v$. This relation is no doubt in reality much more complicated, and its form changes, as $v$ varies, from a form nearly equivalent to (i) to a form nearly equivalent to (ii). But, from a mathematical point of view, there is nothing to prevent us from contemplating an ideal state of things in which, for all values of $v$ not less than a certain value $V$, (i) would be exactly true, and (ii) exactly true for all values of $v$ less than $V$. And then we might regard the two equations as together defining $p$ as a function of $v$. It is an example of a function which for some values of $v$ is defined by one formula and for other values of $v$ is defined by another.

This function possesses the characteristic (2); to any value of $v$ only one value of $p$ corresponds: but it does not possess (1). For $p$ is not defined as a function of $v$ for negative values of $v$; a ‘negative volume’ means nothing, and so negative values of $v$ do not present themselves for consideration at all.

5. Suppose that a perfectly elastic ball is dropped (without rotation) from a height $\frac{1}{2}g{\tau }^2$ on to a fixed horizontal plane, and rebounds continually.

The ordinary formulae of elementary dynamics, with which the reader is probably familiar, show that $h=\frac{1}{2}g{t}^2$ if $0\leqq t\leqq \tau$, if $\tau \leqq t\leqq 3\tau$, and generally

6. Suppose that $y$ is defined as being the largest prime factor of $x$. This is an instance of a definition which only applies to a particular class of values of $x$, viz. integral values. ‘The largest prime factor of $\frac{11}{3}$ or of $\sqrt{2}$ or of $\pi$’ means nothing, and so our defining relation fails to define for such values of $x$ as these. Thus this function does not possess the characteristic (1). It does possess (2), but not (3), as there is no simple formula which expresses $y$ in terms of $x$.

7. Let $y$ be defined as the denominator of $x$ when $x$ is expressed in its lowest terms. This is an example of a function which is defined if and only if $x$ is rational. Thus $y=7$ if $x=-11/7$: but $y$ is not defined for $x=\sqrt{2}$, ‘the denominator of $\sqrt{2}$’ being a meaningless form of words.

8. Let $y$ be defined as the height in inches of policeman $Cx$, in the Metropolitan Police, at 5.30 P.M. on 8 Aug. 1907. Then $y$ is defined for a certain number of integral values of $x$, viz. $1$, $2$, …, $N$, where $N$ is the total number of policemen in division $C$ at that particular moment of time.

21. The graphical representation of functions. Suppose that the variable $y$ is a function of the variable $x$. It will generally be open to us also to regard $x$ as a function of $y$, in virtue of the functional relation between $x$ and $y$. But for the present we shall look at this relation from the first point of view. We shall then call $x$ the independent variable and $y$ the dependent variable; and, when the particular form of the functional relation is not specified, we shall express it by writing

The nature of particular functions may, in very many cases, be illustrated and made easily intelligible as follows. Draw two lines $OX$, $OY$ at right angles to one another and produced indefinitely in both directions. We can represent values of $x$ and $y$ by distances measured from $O$ along the lines $OX$, $OY$ respectively, regard being paid, of course, to sign, and the positive directions of measurement being those indicated by arrows in Fig. 6.

Let $a$ be any value of $x$ for which $y$ is defined and has (let us suppose) the single value $b$. Take $OA=a$, $OB=b$, and complete the rectangle $OAPB$. Imagine the point $P$ marked on the diagram. This marking of the point $P$ may be regarded as showing that the value of $y$ for $x=a$ is $b$.

If to the value $a$ of $x$ correspond several values of $y$ (say $b$, $b\prime$, $b\prime \prime$), we have, instead of the single point $P$, a number of points $P$, $P\prime$, $P\prime \prime$.

We shall call $P$ the point ; $a$ and $b$ the coordinates of $P$ referred to the axes $OX$, $OY$; $a$ the abscissa, $b$ the ordinate of $P$; $OX$ and $OY$ the axis of $x$ and the axis of $y$, or together the axes of coordinates, and $O$ the origin of coordinates, or simply the origin.

Let us now suppose that for all values $a$ of $x$ for which $y$ is defined, the value $b$ (or values $b$, $b\prime$, $b\prime \prime ,\dots$) of $y$, and the corresponding point $P$ (or points $P$, $P\prime$, $P\prime \prime ,\dots$), have been determined. We call the aggregate of all these points the graph of the function $y$.

To take a very simple example, suppose that $y$ is defined as a function of $x$ by the equation

where $A$, $B$, $C$ are any fixed numbers.²⁰ Then $y$ is a function of $x$ which possesses all the characteristics (1), (2), (3) of § 20. It is easy to show that the graph of $y$ is a straight line. The reader is in all probability familiar with one or other of the various proofs of this proposition which are given in text-books of Analytical Geometry.

We shall sometimes use another mode of expression. We shall say that when $x$ and $y$ vary in such a way that equation (1) is always true, the locus of the point  is a straight line, and we shall call (1) the equation of the locus, and say that the equation represents the locus. This use of the terms ‘locus’, ‘equation of the locus’ is quite general, and may be applied whenever the relation between $x$ and $y$ is capable of being represented by an analytical formula.

The equation $Ax+By+C=0$ is the general equation of the first degree, for $Ax+By+C$ is the most general polynomial in $x$ and $y$ which does not involve any terms of degree higher than the first in $x$ and $y$. Hence the general equation of the first degree represents a straight line. It is equally easy to prove the converse proposition that the equation of any straight line is of the first degree.

We may mention a few further examples of interesting geometrical loci defined by equations. An equation of the form

22. Polar coordinates. In what precedes we have determined the position of $P$ by the lengths of its coordinates $OM=x$, $MP=y$. If $OP=r$ and $MOP=\theta$, $\theta$ being an angle between $0$ and $2\pi$ (measured in the positive direction), it is evident that

and that the position of $P$ is equally well determined by a knowledge of $r$ and $\theta$. We call $r$ and $\theta$ the polar coordinates of $P$. The former, it should be observed, is essentially positive.²¹

If $P$ moves on a locus there will be some relation between $r$ and $\theta$, say or . This we call the polar equation of the locus. The polar equation may be deduced from the equation (or vice versa) by means of the formulae above.

Thus the polar equation of a straight line is of the form

23. Further examples of functions and their graphical representation. The examples which follow will give the reader a better notion of the infinite variety of possible types of functions.

A. Polynomials. A polynomial in $x$ is a function of the form

First let $m=2$. Then three points on the graph are , , . Any number of additional points on the graph may be found by assigning other special values to $x$: thus the values

give

If the reader will plot off a fair number of points on the graph, he will be led to conjecture that the form of the graph is something like that shown in Fig. 8. If he draws a curve through the special points which he has proved to lie on the graph and then tests its accuracy by giving $x$ new values, and calculating the corresponding values of $y$, he will find that they lie as near to the curve as it is reasonable to expect, when the inevitable inaccuracies of drawing are considered. The curve is of course a parabola.

There is, however, one fundamental question which we cannot answer adequately at present. The reader has no doubt some notion as to what is meant by a continuous curve, a curve without breaks or jumps; such a curve, in fact, as is roughly represented in Fig. 8. The question is whether the graph of the function $y=x^2$ is in fact such a curve. This cannot be proved by merely constructing any number of isolated points on the curve, although the more such points we construct the more probable it will appear.

This question cannot be discussed properly until Ch. V. In that chapter we shall consider in detail what our common sense idea of continuity really means, and how we can prove that such graphs as the one now considered, and others which we shall consider later on in this chapter, are really continuous curves. For the present the reader may be content to draw his curves as common sense dictates.

It is easy to see that the curve $y=x^2$ is everywhere convex to the axis of $x$. Let $P_0$, $P_1$ (Fig. 8) be the points , . Then the coordinates of a point on the chord $P_0{P}_1$ are $x=\lambda x_0+\mu x_1$, $y=\lambda x_0^2+\mu x_1^2$, where $\lambda$ and $\mu$ are positive numbers whose sum is $1$. And

The curve $y=x^4$ is similar to $y=x^2$ in general appearance, but flatter near $O$, and steeper beyond the points $A$, $A\prime$ (Fig. 9), and $y=x^m$, where $m$ is even and greater than $4$, is still more so. As $m$ gets larger and larger the flatness and steepness grow more and more pronounced, until the curve is practically indistinguishable from the thick line in the figure.

Fig. 9.

Fig. 10.

The reader should next consider the curves given by $y=x^m$, when $m$ is odd. The fundamental difference between the two cases is that whereas when $m$ is even , so that the curve is symmetrical about $OY$, when $m$ is odd , so that $y$ is negative when $x$ is negative. Fig. 10 shows the curves $y=x$, $y=x^3$, and the form to which $y=x^m$ approximates for larger odd values of $m$.

It is now easy to see how (theoretically at any rate) the graph of any polynomial may be constructed. In the first place, from the graph of $y=x^m$ we can at once derive that of $C{x}^m$, where $C$ is a constant, by multiplying the ordinate of every point of the curve by $C$. And if we know the graphs of  and , we can find that of by taking the ordinate of every point to be the sum of the ordinates of the corresponding points on the two original curves.

The drawing of graphs of polynomials is however so much facilitated by the use of more advanced methods, which will be explained later on, that we shall not pursue the subject further here.

Examples XI. 1. Trace the curves $y=7x^4$, $y=3x^5$, $y=x^{10}$.

[The reader should draw the curves carefully, and all three should be drawn in one figure.²² He will then realise how rapidly the higher powers of $x$ increase, as $x$ gets larger and larger, and will see that, in such a polynomial as

2. Compare the relative magnitudes of $x^{12}$, $1,000,000x^6$, $1,000,000,000,000x$ when $x=1$, $10$, $100$, etc.

[The reader should make up a number of examples of this type for himself. This idea of the relative rate of growth of different functions of $x$ is one with which we shall often be concerned in the following chapters.]

3. Draw the graph of $a{x}^2+2bx+c$.

[Here . If we take new axes parallel to the old and passing through the point $x=-b/a$, , the new equation is $y\prime =ax{\prime }^2$. The curve is a parabola.]

4. Trace the curves $y=x^3-3x+1$, , .

24. B. Rational Functions. The class of functions which ranks next to that of polynomials in simplicity and importance is that of rational functions. A rational function is the quotient of one polynomial by another: thus if ,  are polynomials, we may denote the general rational function by

In the particular case when  reduces to unity or any other constant (i.e. does not involve $x$),  reduces to a polynomial: thus the class of rational functions includes that of polynomials as a sub-class. The following points concerning the definition should be noticed.

(1) We usually suppose that  and  have no common factor $x+a$ or $x^p+a{x}^{p-1}+b{x}^{p-2}+\cdots +k$, all such factors being removed by division.

(2) It should however be observed that this removal of common factors does as a rule change the function. Consider for example the function $x/x$, which is a rational function. On removing the common factor $x$ we obtain $1/1=1$. But the original function is not always equal to $1$: it is equal to $1$ only so long as $x\ne 0$. If $x=0$ it takes the form $0/0$, which is meaningless. Thus the function $x/x$ is equal to $1$ if $x\ne 0$ and is undefined when $x=0$. It therefore differs from the function $1$, which is always equal to $1$.

(3) Such a function as

(4) But, as appears from the particular example considered under (3), there will generally be a certain number of values of $x$ for which the function is not defined even when it has been reduced to a rational function of the standard form. These are the values of $x$ (if any) for which the denominator vanishes. Thus is not defined when $x=1$ or $2$.

(5) Generally we agree, in dealing with expressions such as those considered in (2) and (3), to disregard the exceptional values of $x$ for which such processes of simplification as were used there are illegitimate, and to reduce our function to the standard form of rational function. The reader will easily verify that (on this understanding) the sum, product, or quotient of two rational functions may themselves be reduced to rational functions of the standard type. And generally a rational function of a rational function is itself a rational function: i.e. if in , where $P$ and $Q$ are polynomials, we substitute , we obtain on simplification an equation of the form .

(6) It is in no way presupposed in the definition of a rational function that the constants which occur as coefficients should be rational numbers. The word rational has reference solely to the way in which the variable $x$ appears in the function. Thus

The use of the word rational arises as follows. The rational function may be generated from $x$ by a finite number of operations upon $x$, including only multiplication of $x$ by itself or a constant, addition of terms thus obtained and division of one function, obtained by such multiplications and additions, by another. In so far as the variable $x$ is concerned, this procedure is very much like that by which all rational numbers can be obtained from unity, a procedure exemplified in the equation

Again, any function which can be deduced from $x$ by the elementary operations mentioned above using at each stage of the process functions which have already been obtained from $x$ in the same way, can be reduced to the standard type of rational function. The most general kind of function which can be obtained in this way is sufficiently illustrated by the example

25. The drawing of graphs of rational functions, even more than that of polynomials, is immensely facilitated by the use of methods depending upon the differential calculus. We shall therefore content ourselves at present with a very few examples.

Examples XII. 1. Draw the graphs of $y=1/x$, $y=1/x^2$, $y=1/x^3$, ….

[The figures show the graphs of the first two curves. It should be observed that since $1/0$, $1/0^2$, … are meaningless expressions, these functions are not defined for $x=0$.]

Fig. 11.

Fig. 12.

2. Trace , , , and taking various values, positive and negative, for $a$ and $b$.

3. Trace

4. Trace , , where $a<b<c$.

5. Sketch the general form assumed by the curves $y=1/x^m$ as $m$ becomes larger and larger, considering separately the cases in which $m$ is odd or even.

26. C. Explicit Algebraical Functions. The next important class of functions is that of explicit algebraical functions. These are functions which can be generated from $x$ by a finite number of operations such as those used in generating rational functions, together with a finite number of operations of root extraction. Thus

It should be noticed that there is an ambiguity of notation involved in such an equation as $y=\sqrt{x}$. We have, up to the present, regarded (e.g.) $\sqrt{2}$ as denoting the positive square root of $2$, and it would be natural to denote by $\sqrt{x}$, where $x$ is any positive number, the positive square root of $x$, in which case $y=\sqrt{x}$ would be a one-valued function of $x$. It is however often more convenient to regard $\sqrt{x}$ as standing for the two-valued function whose two values are the positive and negative square roots of $x$.

The reader will observe that, when this course is adopted, the function $\sqrt{x}$ differs fundamentally from rational functions in two respects. In the first place a rational function is always defined for all values of $x$ with a certain number of isolated exceptions. But $\sqrt{x}$ is undefined for a whole range of values of $x$ (i.e. all negative values). Secondly the function, when $x$ has a value for which it is defined, has generally two values of opposite signs.

The function $\sqrt[3]{x}$, on the other hand, is one-valued and defined for all values of $x$.

Examples XIII. 1. , where $a<b$, is defined only for $a\leqq x\leqq b$. If $a<x<b$ it has two values: if $x=a$ or $b$ only one, viz. $0$.

2. Consider similarly

3. Trace the curves $y^2=x$, $y^3=x$, $y^2=x^3$.

4. Draw the graphs of the functions

27. D. Implicit Algebraical Functions. It is easy to verify that if

where $R_1$, $R_2$, …, $R_m$ are rational functions of $x$: and the reader will easily verify that, if $y$ is any one of the functions considered in the last set of examples, $y$ satisfies an equation of this form. It is naturally suggested that the same is true of any explicit algebraic function. And this is in fact true, and indeed not difficult to prove, though we shall not delay to write out a formal proof here. An example should make clear to the reader the lines on which such a proof would proceed. Let

and we have only to eliminate $u$, $v$, $w$ between these equations in order to obtain an equation of the form desired.

We are therefore led to give the following definition: a function will be said to be an algebraical function of $x$ if it is the root of an equation such as (1), i.e. the root of an equation of the $m$^th degree in $y$, whose coefficients are rational functions of $x$. There is plainly no loss of generality in supposing the first coefficient to be unity.

This class of functions includes all the explicit algebraical functions considered in § 26. But it also includes other functions which cannot be expressed as explicit algebraical functions. For it is known that in general such an equation as (1) cannot be solved explicitly for $y$ in terms of $x$, when $m$ is greater than $4$, though such a solution is always possible if $m=1$, $2$, $3$, or $4$ and in special cases for higher values of $m$.

The definition of an algebraical function should be compared with that of an algebraical number given in the last chapter (Misc. Exs. 32).

Examples XIV. 1. If $m=1$, $y$ is a rational function.

2. If $m=2$, the equation is $y^2+R_{1}y+R_2=0$, so that

If $m=3$ or $4$, we can use the methods explained in treatises on Algebra for the solution of cubic and biquadratic equations. But as a rule the process is complicated and the results inconvenient in form, and we can generally study the properties of the function better by means of the original equation.

3. Consider the functions defined by the equations

4. Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions

5. Consider the equation $y^4=x^2$.

[Here $y^2=\pm x$. If $x$ is positive, $y=\sqrt{x}$: if negative, $y=\sqrt[]{-x}$. Thus the function has two values for all values of $x$ save $x=0$.]

6. An algebraical function of an algebraical function of $x$ is itself an algebraical function of $x$.

[For we have

Here all the capital letters denote rational functions.]

7. An example should perhaps be given of an algebraical function which cannot be expressed in an explicit algebraical form. Such an example is the function $y$ defined by the equation

28. Transcendental functions. All functions of $x$ which are not rational or even algebraical are called transcendental functions. This class of functions, being defined in so purely negative a manner, naturally includes an infinite variety of whole kinds of functions of varying degrees of simplicity and importance. Among these we can at present distinguish two kinds which are particularly interesting.

E. The direct and inverse trigonometrical or circular functions. These are the sine and cosine functions of elementary trigonometry, and their inverses, and the functions derived from them. We may assume provisionally that the reader is familiar with their most important properties.²³

Examples XV. 1. Draw the graphs of $cosx$, $sinx$, and $acosx+bsinx$.

[Since , where $\beta =\sqrt[]{a^2+b^2}$, and $\alpha$ is an angle whose cosine and sine are $a/\sqrt[]{a^2+b^2}$ and $b/\sqrt[]{a^2+b^2}$, the graphs of these three functions are similar in character.]

2. Draw the graphs of ${cos}^{2}x$, ${sin}^{2}x$, $a{cos}^{2}x+b{sin}^{2}x$.

3. Suppose the graphs of  and  drawn. Then the graph of

4. Show that the graph of $cospx+cosqx$ lies between those of and , touching each in turn. Sketch the graph when is small.

5. Draw the graphs of $x+sinx$, , $xsinx$, .

6. Draw the graph of .

[If , then $y=0$ when $x=1/m\pi$, where $m$ is any integer. Similarly $y=1$ when and $y=-1$ when . The curve is entirely comprised between the lines $y=-1$ and $y=1$ (Fig. 13). It oscillates up and down, the rapidity of the oscillations becoming greater and greater as $x$ approaches $0$. For $x=0$ the function is undefined. When $x$ is large $y$ is small.²⁴ The negative half of the curve is similar in character to the positive half.]

7. Draw the graph of .

[This curve is comprised between the lines $y=-x$ and $y=x$ just as the last curve is comprised between the lines $y=-1$ and $y=1$ (Fig. 14).]

Fig. 13.

Fig. 14.

8. Draw the graphs of , , , , , , .

9. Draw the graphs of $cos{x}^2$, $sin{x}^2$, $acos{x}^2+bsin{x}^2$.

10. Draw the graphs of