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Composition and Invertible Mappings Homework

Directions: Write your homework legibly with a pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution/proof and show all steps. Do not turn in your scratch work. Staple your pages together, with the problems in the correct order, and use this page as a cover sheet with your name at the top right.

(1) Let and be mappings from to Write a formula for and In each case determine the image.

(2) Prove that if then and

(3) Given the functions defined by and defined by If possible, determine their inverse functions and write a formula for each one.

(4) Assume that and Prove or disprove the following statement, its converse, and its contrapositive: If is one-to-one and is onto, then is one-to-one and onto.

(5) Assume that and Prove or disprove the following statement, its converse, and its contrapositive: If is not one-to-one, then is not invertible.

(6) Prove that the inverse of an invertible mapping is invertible.

(7) Prove that if is onto, and then Show that is onto is a necessary condition.

(8) Prove that if is one-to-one, and then Show that is one-to-one is a necessary condition.

(9) Suppose that and are one-to-one and onto, then

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