# Rajat Gupta Essay

757 Words Sep 25th, 2014 4 Pages
Module 3 Written Assignment: Part 3

General Expectations: (i). For full credit, your written assignments must be accompanied by a narrative explanation/rationale for the process that you used to solve each problem. How did you choose the steps? What is the logic behind the choices that you made? Explain why the problems were solved the way they were solved. Use complete sentences, good English, and proper mathematical notation. (ii). For full credit, your written assignments must include the statement of each problem so the reader knows what you are trying to demonstrate. In cases where the assignment refers you to a book problem, you must also copy the statement of the appropriate problem in the book. Note: Be sure to do the practice
bijective (d). neither injective nor surjective. In each case, explain how your example satisﬁes the given conditions. 7. Deﬁne f : R → R such that f (x) = x . This is the ceiling function and assigns to the real number x the smallest integer that is greater than or equal to x. For example 1.2 = 2 because 2 is the smallest integer that is greater than or equal to 1.2. Similarly, we can compute 1 = 1 and 0.9 = 1. (a.) Is f one-to-one? Explain. (b.) Is f onto? Explain. (c.) How would your answer to part (b.) change if we deﬁned the function as follows: f : R → Z such that f (x) = x ?

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Section 3.2: Practice: Do # 7a from Section 3.2. Check your answer in the back of the book. 1. Show that the following function is one-to-one. Find the range of the function and a suitable inverse 1 for f : A → R where A = {x ∈ R| x = 2} and f (x) = x−2 + 3. 2. Prove that the function f : Q → Q given by f (x) = 3x + 9 and the function g : Q → Q given by g(y) = y − 3 are inverses of each other. [Use the ﬁll-in-the-blanks proof applet to help you]. 3 3. Determine whether each of the given functions is a bijection from R to itself. Justify your answers. (a.) f (x) = x3 + 1 (b.) f (x) = 2x − 9 4. Let S = {1, 2, 3, 4, 5} and let T = {3, 4, 5, 6, 7}. Deﬁne functions f : S → T and g : S → S as follows: f = {(2, 6), (1, 6), (3, 4), (5, 3), (4, 5)} and g = {(2, 3), (1, 2), (4, 5), (5, 1), (3, 4)}. You may ﬁnd the video: http://youtu.be/eNIIy-wA5xc helpful. (a.) Find f ◦ g or explain

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