# Rajat Gupta Essay

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

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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