H2 - CS123 Youssef February 2 2010 Homework 2 Due Date...

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CS123 February 2, 2010 Youssef Homework 2 Due Date: February 25, 2009 Problem 1: (20 points) Let R denote the set of real numbers, and Z the set of integers. Let also f : R R , g : R R , and h : Z R be 3 functions defined as follows: f ( x ) = 8 x - 3 g ( x ) = 3 x 2 + 4 h ( x ) = 4 x +3 2 . a) Is f one-to-one? Onto? Prove your answer. If f is one-to-one and onto, find f - 1 . b) Calculate g (0) , g (1) , g (2) . c) Given two sets E and F , and a function u : E F , and for every y F , define u ( y ) to be the following set: u ( y ) = { x E | u ( x ) = y } . Determine g (0) , g (4) , g (16) , g ( - 1). d) Is g one-to-one? Onto? Prove your answer. e) Is h one-to-one? Onto? Prove your answer. f) Assume now that h : R R but h has otherwise the same definition. Calculate g f ( x ) , h g ( x ) , h ( g f )( x ) , ( h g ) f ( x ). Problem 2: (20 points) Let N be the set of natural numbers, and Z the set of integers. Let f : N Z and g : N Z be two functions defined recursively as follows: f (0) = 2 and f ( n ) = 5 f ( n - 1) - 4 for n 1 g (0) = - 1, g (1) = 1 and g ( n ) = 3 2 g ( n - 1) + 9 2 g ( n - 2) + 10 for n 2.
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