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HW2 - a 2 3 to be the number 3 √ a 2 Prove that this...

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Math 23b Homework 2 Spring, 2009 Due Wednesday, February 4. Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/, w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the problem does not explicitly ask you to do so. 1. Problem 1.45 Page 24. For each deﬁnition, check whether or not the function is well- deﬁned. 2. Problem 1.49 Page 24. Recall that f is bounded if there is a real number M such that | f ( x ) | ≤ M for all x R . In part (c) consider the possibility that the values of f + g are small while the values of f and g are large and cancel each other when you add them. Something similar happens in (d). 3. Let P be the set of positive numbers, and B be the set of numbers x such that 0 < x < 1. Deﬁne f : P B , f ( x ) = 1 / (1 + x 2 ). Prove that f is a bijection by using the Bijectivity Test. (You should ﬁnd the inverse of f ﬁrst.) 4. Problem 2.1 Page 44. 5. Fractional powers. For positive a R , we deﬁne the symbol
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Unformatted text preview: a 2 / 3 to be the number 3 √ a 2 . Prove that this number is equal to ( 3 √ a ) 2 . (Hint: By the Cubic Root Theorem, this number is the unique solution to the equation x 3 = a 2 .) Let p,q to be positive integers such that p/q = 2 / 3. Deﬁne the symbol a p/q to be the number q √ a p . Prove that this number is also equal to 3 √ a 2 . 6. The Fractional Power Law. Do this one for fun. Do not hand in your proof but feel free to discuss it with me. Let p,q,r,s be positive integers. For positive a ∈ R , we deﬁne the symbol a p/q to be the number q √ a p . Prove that a p/q a r/s = a p/q + r/s and that ( a p/q ) r/s = a ( pr ) / ( qs ) . (Hint: Consider the equations x qs = a ps + qr and x qs = a pr respectively and use the n th Root Theorem.) You can also prove that if p/q = r/s , then q √ a p = s √ a r by a similar consideration....
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