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 $$P(x)$$$$Q(x)$$ are polynomials, we may denote the general rational function by $R(x) = \frac{P(x)}{Q(x)}.$

In the particular case when $$Q(x)$$ reduces to unity or any other constant ( does not involve $$x$$), $$R(x)$$ 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 $$P(x)$$ and $$Q(x)$$ have no common factor $$x + a$$ or $$x^{p} + ax^{p-1} + bx^{p-2} + \dots + k$$, 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 $$x\neq 0$$. If $$x = 0$$ it takes the form $$0/0$$, which is meaningless. Thus the function $$x/x$$ is equal to $$1$$ if $$x\neq 0$$ and is undefined when $$x = 0$$. It therefore differs from the function $$1$$, which is always equal to $$1$$.

(3) Such a function as $\left(\frac{1}{x + 1} + \frac{1}{x – 1}\right) \bigg/ \left(\frac{1}{x} + \frac{1}{x – 2}\right)$ may be reduced, by the ordinary rules of algebra, to the form $\frac{x^{2}(x – 2)}{(x – 1)^{2} (x + 1)},$ which is a rational function of the standard form. But here again it must be noticed that the reduction is not always legitimate. In order to calculate the value of a function for a given value of $$x$$ we must substitute the value for $$x$$ in the function in the form in which it is given. In the case of this function the values $$x = -1$$, $$1$$$$0$$$$2$$ all lead to a meaningless expression, and so the function is not defined for these values. The same is true of the reduced form, so far as the values $$-1$$ and $$1$$ are concerned. But $$x = 0$$ and $$x = 2$$ give the value $$0$$. Thus once more the two functions are not the same.

(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 $$(x^{2} – 7)/(x^{2} – 3x + 2)$$ 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:  if in $$z = P(y)/Q(y)$$, where $$P$$ and $$Q$$ are polynomials, we substitute $$y = P_{1}(x)/Q_{1}(x)$$, we obtain on simplification an equation of the form $$z = P_{2}(x)/Q_{2}(x)$$.

(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 $\frac{x^{2} + x + \sqrt{3}}{x\sqrt[3]{2} – \pi}$ is a rational function.

The use of the word rational arises as follows. The rational function $$P(x)/Q(x)$$ 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 $\frac{5}{3} = \frac{1 + 1 + 1 + 1 + 1}{1 + 1 + 1}.$

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 $\Biggl(\frac{x}{x^{2} + 1} + \frac{2x + 7}{x^{2} + \dfrac{11x – 3\sqrt{2}}{9x + 1}}\Biggr) \Bigg/ \left(17 + \frac{2}{x^{3}}\right),$ which can obviously be reduced to the standard type of rational function.

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.

Example 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 $$x = 0$$.]

2. Trace $$y = x + (1/x)$$, $$x – (1/x)$$, $$x^{2} + (1/x^{2})$$, $$x^{2} – (1/x^{2})$$ and $$ax + (b/x)$$ taking various values, positive and negative, for $$a$$ and $$b$$.

3. Trace $y = \frac{x + 1}{x – 1},\quad \left(\frac{x + 1}{x – 1}\right)^{2},\quad \frac{1}{(x – 1)^{2}},\quad \frac{x^{2} + 1}{x^{2} – 1}.$

4. Trace $$y = 1/(x – a)(x – b)$$, $$1/(x – a)(x – b)(x – c)$$, 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.