Relations and Functions

Mathematics often pairs objects, such as a student with a roll number or a number with its square. This chapter makes such pairings precise. It begins with ordered pairs, in which order matters, so (a, b) = (c, d) only when a = c and b = d. The Cartesian product A x B of two non-empty sets is the set of all ordered pairs (a, b) with a in A and b in B, and if n(A) = p and n(B) = q then n(A x B) = pq. A relation from A to B is any subset of A x B, with a domain, codomain and range, and there are 2^(pq) relations from A to B. A function is a special relation in which every element of A has exactly one image in B. You will study real-valued functions and their graphs, including the identity, constant, polynomial, rational, modulus, signum and greatest integer functions. You will find the domain and range of functions given by formulas, and learn the algebra of real functions.

Detailed Worksheet: Relations and Functions Section A - Definitions (10 marks) 1. Find x and y if (x + 1, y - 2) = (3, 1). (2 marks) 2. If A = {1, 2} and B = {3, 4, 5}, write A x B. (2 marks) 3. If n(A) = 3 and n(B) = 2, find the number of relations from A to B. (2 marks) 4. Find [2.7] and [-1.5], where [x] is the greatest integer function. (2 marks) 5. Define a function as a special kind of relation. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the domain of the function f(x) = (x^2 + 3x + 5)/(x^2 - 5x + 4). (3 marks) 7. Find the domain and range of the relation R = {(x, x + 5) : x in {0, 1, 2, 3, 4, 5}}. (3 marks) 8. If f(x) = x + 1 and g(x) = 2x - 3, find (f + g)(x), (fg)(x) and (f/g)(x). (3 marks) 9. Determine whether each relation is a function: (i) {(2, 1), (5, 1), (8, 1), (11, 1)} (ii) {(2, 1), (2, 3), (4, 5)}. Give reasons. (3 marks) 10. A x A has 9 elements, two of which are (-1, 0) and (0, 1). Find A and the remaining elements. (3 marks) Section C - Diagrams (10 marks) 11. Draw the graph of the modulus function f(x) = mod(x) and state its domain and range. (4 marks) 12. Draw the graph of the signum function and state its range. (3 marks) 13. Draw an arrow diagram for the relation R = {(a, b) : b = a + 1} on A = {1, 2, 3, 4, 5, 6} and state its domain and range. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Define the greatest integer function. Draw its graph for -2 <= x < 3 and state its domain and range. (5 marks) 15. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find (i) A x (B intersection C) (ii) (A x B) intersection (A x C) (iii) A x (B union C). (5 marks) 16. Find the domain and range of (i) f(x) = x^2 + 2 (ii) f(x) = sqrt(9 - x^2) (iii) f(x) = 1/(x - 3). (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write sets in roster form where asked. mod(x) denotes the modulus of x.