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) = x2 , 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 x2 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. 30 X (k + 3)2 k=2 (8) Evaluate the sum and simplify as much as possible. Leave large exponential expressions of the form ab with aj, b integers in that form in your final answer and do not evaluate them. 9000 2n1 X 7 62n+1 k=10