By Frédéric Cao

ISBN-10: 3540004025

ISBN-13: 9783540004028

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Moreover, if / and g are real monotone functions of real variable, gof too will be monotone, or better: g o f is increasing if b o t h / and g are either increasing or decreasing, and decreasing otherwise. Let us prove only one of these properties. Let for example / increase and g decrease; if xi < X2 are elements in d o m ^ o / , the monotone behaviour of / implies f{xi) < /(X2); now the monotonicity of g yields g{f{xi)) > g{f[x2)), so g o f is decreasing. We observe finally t h a t if / is a one-to-one function (and as such it admits inverse / ~ ^ ) , then / " ' o fix) = r\f{x)) =x, Vx e dom / , forHy) = firHy)) = y, Vyeim/.

22. Important values of these maps are listed in the following table (where k is any integer): sina: = 0 for x = fcTr, sin x = 1 for X — -^ + 2k7r, TV s i n x = —1 for x = ——-f2fc7r, COS X = 0 TT for X = ~ 4- fcTT, cos X = 1 for X = 2fc7r, c o s x = —1 for X = TT -f 2fc7r. 22. Graph of the map y = cosx 27r 54 2 Functions Concerning monotonicity, one has y = smx strictly increasing on - - + 2fc7r, - + 2/c7r strictly decreasing on - + 2/c7r, —- + 2k7r IS < ^ ^, 37r ^, • strictly decreasing on [2fc7r, TT + 2A:7r] y = cos X IS strictly increasing on [TT + 2A:7r, 27r + 2/c7r].

28 1 Basic notions b) 5 = ( - o c , - 2 ) U ( 2 , 5 ) . c) We can write x^ - 5 x + 4 _ ( x - 4 ) ( x - 1) ^2-9 ~ ( J : - 3 ) ( X + 3) ' whence the first set is (—3,1) U (3,4). To find the second set, let us solve the irrational equation \Jlx + 1 + x = 17, which we write as \/lx + 1 = 17—x. The radicand must necessarily be positive, hence x > — y. , X < 17. Thus for —y < x < 17, squaring yields 7x + 1 = (17 - xf , x^ - 41x + 288 = 0. The latter equation has two solutions xi = 9, X2 = 32 (which fails the constraint X < 17, and as such cannot be considered).

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Mathematical Analysis I by Frédéric Cao


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