By Saxelby F.M.
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The typical scientific calculator has many of the most common functions built in (via the 'function keys'). Such a calculator illustrates the need for using a 'suitable input'—if we enter —1 and then press the y/ key, we get an error message. The range of suitable inputs will, of course, vary with the choice of function. The chapter begins by discussing the general case of a function from one general set to another, introducing the language associated with functions and the ideas of composition.
11 Let η € Ν. Show that as η increases x" decreases for χ € (0,1) and increases for χ e (1, oo). S o l u t i o n Let η e N. In both cases χ > 0 so that x" > 0. e. x < x". Similarly, when χ G (Ι,οο), χ > 1 and hence n + 1 X n + 1 > χ". 1. 1 Let a,d € Ζ with d φ 0- We say that d divides a if a = dk for some k € Z. When d divides ο we write d | α and call d a divisor or factor of a. Otherwise we write d \ a. For example, 20 = 4 χ 5 20 = (-10) χ (-2) so so 20 φ 3k for any integer k 4 I 20, - 10 ] 20, so 3 f 20.
2 But 0 < s i < 2 and 0 < s < 2. So s + s = 0 means that s = s = 0. Therefore 2 l = 2 / ( S 2 ) t =>· «Ι = 2 52· Hence, / is injective. (2) Let t € [1,5]. We want s € [0,2] with / ( s ) = t. e. s = ±\/i — 1. For s Ε [0,2] we must take the non-negative square root. Therefore, define s = \/t — 1. Since 2 1<<<5 => 0 < * - l < 4 => 0 < s < 2 , we get s € [0,2]. Further, f{s) = s 2 + 1 = (t - 1) + 1 = t. Hence, / is surjective. (3) Since / is injective and surjective, it is bijective. 5 43 Remark For real functions, injectiveness, surjectiveness and bijectiveness can be inter preted geometrically as follows.