1) Find the range of these functions. Give your answer as a simple/short sentence using plain English. The following solution to (a): S = {2x | x Z} despite being correct, will receive 0 points. Recall that N is the set of all natural numbers numbers {0, 1, 2, 3, . . .}, and Z+ is the set of positive integers {1, 2, 3, . . .}. (a) f : Z R, f (x) = 2x. (b) f : N R, f (x) = 2x. (c) f : R R, f (x) = 2x. (d) f : N R, f (x) = last digit of 2x. (e) f : N Z+ R, f (x, y) = x/y. Remember to simplify your answer. For example, the answer {x/y|x N, y Z+} to part (d), despite being correct will get 0 marks

. 2) (15pts) For each of the following functions from real numbers to real numbers prove or disprove the following: The function is injective. The function is surjective. The function is bijective. If function is a bijection, then find its inverse

(a) f (x) = 2bxc + 1 (b) f (x) = 10x + 7 (c) f (x) = |6x + 3| (d) f (x) = 5x2 1 (e) f (x) = 2x5 3

3) Let f and g both be functions from real numbers to real numbers. Let f (x) = x2 5 and g(x) = 3 + 2x. Define each of the following. Simplify your answer and show your work. (a) f g (b) g f (c) (g f ) g

4) Let g : A B and f : B C be two functions. Prove or disprove the followings: (a) if f and g are surjective, then (f g) is surjective. (b) if f and g are injective, then (f g) is injective.