December 1998
MA214: DISCRETE MATHEMATICS

QUESTION 4

Total Marks: 20 Marks

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SUGGESTED SOLUTIONS
for 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]
(ii) Calculate the inverse of function f(x). [1]
(iii) Calculate the inverse of function g(x). [3]
(iv) Hence, or otherwise, calculate (g o f)-1(x)

 

 [3]
(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 bigequ.GIF (60 bytes) y

 
(i) List all the elements of the Cartesian product set X x Y. [2]
(ii) Are the relations R and S functions? Justify your answers.

 

 [2]
(c) All non-negative real numbers can be expressed in the form

x = n + r

where n elementsign.gif (62 bytes)N and 0 smallequ.gif (60 bytes) r < 1.

The function Round(x) of the type realnum.gif (71 bytes)+arrowtoright.gif (56 bytes) realnum.gif (71 bytes)+ is defined as follows:

equation1.gif (475 bytes)

 
(i) Sketch a graph of Round(x). [3]
(ii) Decide whether or not Round is injective, surjective, bijective, justifying your answers. [4]