Dec 1998 MA214: Discrete Mathematics (Question 4)

(a)	Let f and g be the functions f(x) = x + 3 g(x) = 2x - 8
	(i) Give the conditions and explain precisely what it means for a function to be invertible.	[2]
	1 mark for saying that a function is invertible if its inverse relation is also a function. 1 mark for saying that a function is invertible precisely when it is a bijection.
	(ii) Calculate the inverse of function f(x).	[1]
	f^-1(x) = x - 3
	(iii) Calculate the inverse of function g(x).	[3]
	Let y = g(x) y = 2x - 8 ® y + 8 = 2x ® x = 0.5y + 2x ® g^-1 (y) = 0.5y + 4 ® g^-1 (y) = 0.5x + 4 1 mark for calculating the correct inverse. 2 marks for the method - one for knowing to set y equal to g(x) and one for rearranging to make x the subject.
	(iv) Hence, or otherwise, calculate (g o f)^-1(x)	[3]
	There are various methods for solving this problem. Any method which gets the correct answer gets full marks. Mark allocation for two possibilities are given below. Use your judgement based on these for other methods. The shortest route is observing that (g o f)^-1(x) = f g (x) (1 mark) and hence deducing that (g o f)^-1(x) = (0.5x + 4) - 3 (1 mark) and hence getting the correct answer (g o f)^-1(x) = 0.5x + 1 (1 mark) Alternatively, they may expand (g o f)^-1(x) to get (g o f)^-1(x) = 2(x + 3) - 8 = 2x + 2 (1 mark) and then set y = (g o f)^-1(x) and rearrange to get x = 0.5y + 1 (1 mark) and hence conclude that (g o f)^-1(x) = 0.5x + 1 (1 mark)
(b)	Let X and Y be the sets X = {1,2} Y = {1,2,3} and let R and S be the relations mapping X onto Y defined below: xRy when x > y xSy when x y
	(i) List all the elements of the Cartesian product set X x Y.	[2]
	X x Y = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}. Subtract one mark for each error or omission up to a maximum of two.
	(ii) Are the relations R and S functions? Justify your answers.	[2]
	1 mark for observing that R has no two pairs with the same first element and so is a function. *1 mark for observing that S has two pairs with the same first element i.e., (2,1) and (2,2) and so is not* a function.**
(c)	All non-negative real numbers can be expressed in the form x = n + r where n N and 0 r < 1. The function Round(x) of the type ⁺ ⁺is defined as follows:
	(i) Sketch a graph of Round(x).	[3]
	One mark for having a function with steps. One mark for including some labelling of axes indicating that the steps are in the right places. One mark for indicating that the begining of each step is included but the end is not.
	(ii) Decide whether or not Round is injective, surjective, bijective, justifying your answers.	[4]
	One mark for saying that it is not injective because some points in the codomain are mapped onto by more than one point in the domain e.g., 1.2 and 1.4 both map onto 1. One mark for saying that it is not surjective because not all points in the codomain are mapped onto e.g. all positive numbers which are integers. Two marks for saying that it is not bijective because it is neither injective nor surjective.