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 exercises in this assignment. They have been carefully chosen to help you understand the material needed to do the assigned exercises that immediately follow.

Section 3.1: Practice: Do # 1a and 1c from Section 3.1. Check your answers in the back of the book. 1. Let f = {(Mindy, Subaru), (Jasmine, Toyota), (Sharon, VW), (Roger, Kia), (Jose, VW)}. (a.) What is the domain of this relation? (b.) What is the range? (c.) Is this a function? Why or why not? (d.) Is this relation one-to-one? Why or why not? (d.) Define g as {(Mindy, Subaru), (Jasmine, Toyota), (Sharon, VW), (Roger, Kia), (Jose, Oldsmobile)}. Is g a one-to-one function? Why or why not? 2. Prove that the function f : R → R given by f (x) = 4x − 3 is surjective. [You may use the Proof Shuffler applet to help you]. 3. Prove that the function f : [0, ∞) → [4, ∞) given by f (x) = x2 + 4 is bijective. [You may use the Proof Shuffler applet to help you]. 4. Let f = {(2, 1), (2, 2), (2, 3), (4, 4)}. Is this a function? Why or why not? 5. Define f to be a function from R to R by f (x) = x2 + 1. (a.) Is f injective? Explain your answer. (b.) Is f surjective? Explain your answer. 6. Give an example of a function f : Z → Z that is (a). injective but not surjective (b). surjective but not injective (c). 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 why f ◦ g is not defined. Repeat for g ◦ f , f ◦ f and g ◦ g. (b.) Which (if any) of f , g are one-to-one? Which (if any) are onto? Explain. (c.) Find f −1 if it exists. If it doesn’t, explain why not. (d.) Find g −1 if it exists. If it doesn’t, explain why not. Practice: Do # 23a and # 24b from Section 3.2. Check your answers in the back of the book.

5. Do # 23b and #24a from Section 3.2. Extra Credit: Do # 23c or # 24c from Section 3.2.

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