MAT 243 HW 3 (1) (5 pts) Fill in the blank in the statements below: (a) A function f : A → B is one-to-one if and only if (b) A function f : A → B is not one-to-one if and only if (c) A function f : A → B is onto if and only if (d) A function f : A → B is not onto if and only if (e) A function f : A → B is

MAT 243 HW 3 (1) (5 pts) Fill in the blank in the statements below: (a) A function f : A → B is one-to-one if and only if (b) A function f : A → B is not one-to-one if and only if (c) A function f : A → B is onto if and only if (d) A function f : A → B is not onto if and only if (e) A function f : A → B is increasing if and only if (f) A function f : A → B is not increasing if and only if (g) A function f : A → B is decreasing if and only if (h) A function f : A → B is not decreasing if and only if (e) A sequence is a function whose domain (e) An arithmetic sequence is a function whose domain (e) A geometric sequence is a function whose domain (2) Prove or disprove: S = [1, 3) ∩ (2, 3] is the empty set. (3) Let f : [−2, 2] → [−4, 4]; f(x) = x 2 , find (a) f −1 ((0, −4)) (b) f(f −1 ({−1})) (c) f −1 (f({2})) (d) Is it always true that if X is a subset of the domain of a function f then f −1 (f(X))? (4) Prove that f : Z → Z; f(n) = 3n − 5 is one-to-one but not onto. 1 2 MAT 243 HW 3 (5) Prove that g : R → R + ∪ {0}; g(x) = (x + 3)2 is not one-to-one but it is onto. (6) Letf : R → Z; f(x) = b x 2 c. Find f −1 ([−3, 2]) = (7) Evaluate the sum and simplify as much as possible. Show all your work. Calculator answers will not be accepted. X 30 k=2 (k + 3)2 (8) Evaluate the sum and simplify as much as possible. Leave large exponential expressions of the form a b with aj, b integers in that form in your final answer and do not evaluate them. X 9000 k=10 7 2n−1 6 2n+1

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