The curve is entirely comprised between the lines y 1

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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. * The negative half of the curve is similar in character to the positive half.] 7. Draw the graph of x sin(1 /x ). [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 ).] 8. Draw the graphs of x 2 sin(1 /x ), (1 /x ) sin(1 /x ), sin 2 (1 /x ), { x sin(1 /x ) } 2 , a cos 2 (1 /x ) + b sin 2 (1 /x ), sin x + sin(1 /x ), sin x sin(1 /x ). 9. Draw the graphs of cos x 2 , sin x 2 , a cos x 2 + b sin x 2 . 10. Draw the graphs of arc cos x and arc sin x . [If y = arc cos x , x = cos y . This enables us to draw the graph of x , considered as a function of y , and the same curve shows y as a function of x . It is clear that y is only defined for - 1 5 x 5 1, and is infinitely many-valued for these values of x . As the reader no doubt remembers, there is, when - 1 < x < 1, a * See Chs. IV and V for explanations as to the precise meaning of this phrase.
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[II : 28] FUNCTIONS OF REAL VARIABLES 62 Fig. 13. Fig. 14. value of y between 0 and π , say α , and the other values of y are given by the formula 2 ± α , where n is any integer, positive or negative.] 11. Draw the graphs of tan x, cot x, sec x, cosec x, tan 2 x, cot 2 x, sec 2 x, cosec 2 x. 12. Draw the graphs of arc tan x , arc cot x , arc sec x , arc cosec x . Give for- mulae (as in Ex. 10) expressing all the values of each of these functions in terms of any particular value. 13. Draw the graphs of tan(1 /x ), cot(1 /x ), sec(1 /x ), cosec(1 /x ). 14. Show that cos x and sin x are not rational functions of x . [A function is said to be periodic , with period a , if f ( x ) = f ( x + a ) for all values of x for which f ( x ) is defined. Thus cos x and sin x have the period 2 π . It is easy to see that no periodic function can be a rational function, unless it is a constant. For suppose that f ( x ) = P ( x ) /Q ( x ) , where P and Q are polynomials, and that f ( x ) = f ( x + a ), each of these equations holding for all values of x . Let f (0) = k . Then the equation P ( x ) - kQ ( x ) = 0 is satisfied by an infinite number of values of x , viz. x = 0, a , 2 a , etc., and therefore for all values of x . Thus f ( x ) = k for all values of x , i.e. f ( x ) is a constant.] 15. Show, more generally, that no function with a period can be an alge- braical function of x .
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[II : 29] FUNCTIONS OF REAL VARIABLES 63 [Let the equation which defines the algebraical function be y m + R 1 y m - 1 + · · · + R m = 0 (1) where R 1 , . . . are rational functions of x . This may be put in the form P 0 y m + P 1 y m - 1 + · · · + P m = 0 , where P 0 , P 1 , . . . are polynomials in x . Arguing as above, we see that P 0 k m + P 1 k m - 1 + · · · + P m = 0 for all values of x . Hence y = k satisfies the equation (1) for all values of x , and one set of values of our algebraical function reduces to a constant.
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