Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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4.12 (a) False. One counterexample is f ( x ) = e - x . (b) False. Take f ( x ) = 0. (Any constant function will work.) (c) False. Consider f ( x ) = x if 0 6 = x 6 = 1 , 1 if x = 0 , 0 if x = 1 . (d) True. If | f ( x ) | ≤ M for every x R , then M + 1 is not in the image of f . (e) False. Take f ( x ) = x 2 . 4.13 We assume without loss of generality that n > rev( n ) so that a > c . Then x = n - rev( n ) = (100 a + 10 b + c ) - (100 c + 10 b + a ) = (100 a + 10( b - 1) + (10 + c )) - (100 c + 10 b + a ) = (100( a - 1) + 10(10 + b - 1) + (10 + c )) - (100 c + 10 b + a ) = 100(( a - 1) - c ) + 10((9 + b ) - b ) + ((10 + c ) - a ) = 100(( a - c ) - 1) + 10 · 9 + (10 - ( a - c )) . (Note that these algebraic manipulations are motivated by the notion of “borrowing” as in the standard algorithm for subtraction from grade school.) Since 0 c < a 9, we have 1 a - c 9, which implies that 0 ( a - c ) - 1 8 and 1 10 - ( a - c ) 9. Hence the digits of
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HW9solns - 4.12 (a) False. One counterexample is f (x) =...

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