Ordered Pairs
An ordered pair (a, b) is a pair of objects where the order matters. (a, b) = (c, d) if and only if a = c and b = d.
The first component is called the abscissa and the second is called the ordinate.
Cartesian Product
The Cartesian product of sets A and B, denoted by A x B, is the set of all ordered pairs (a, b) where a in A and b in B.
A x B = {(a, b) : a in A, b in B}
If A has m elements and B has n elements, then A x B has mn elements.
Example: If A = {1, 2} and B = {3, 4, 5}, then A x B = {(1,3), (1,4), (1,5), (2,3), (2,4), (2,5)}.
Properties of Cartesian Product
A x B != B x A(unless A = B or either is empty)A x (B cup C) = (A x B) cup (A x C)A x (B cap C) = (A x B) cap (A x C)A x phi = phi x A = phi
Relations
A relation R from set A to set B is a subset of A x B. If (a, b) in R, we say a is related to b, written as a R b.
Domain, Codomain, and Range
- Domain: Set of all first components of ordered pairs in R.
- Codomain: Set B (in a relation from A to B).
- Range: Set of all second components of ordered pairs in R.
Example: For A = {1, 2, 3, 4} and B = {1, 4, 9, 16, 25}, define R = {(x, y) : y = x^2, x in A, y in B}.
Then R = {(1,1), (2,4), (3,9), (4,16)}.
Domain = {1, 2, 3, 4}, Codomain = {1, 4, 9, 16, 25}, Range = {1, 4, 9, 16}.
Types of Relations
- Empty Relation: No element of A is related to any element of A.
- Universal Relation: Every element of A is related to every element of A.
- Reflexive Relation:
(a, a) in Rfor everya in A. - Symmetric Relation: If
(a, b) in R, then(b, a) in R. - Transitive Relation: If
(a, b) in Rand(b, c) in R, then(a, c) in R. - Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
Functions
A relation f from A to B is called a function if every element of A has a unique image in B. Denoted by f : A -> B.
For each x in A, there exists a unique y in B such that f(x) = y.
Domain, Codomain, and Range of a Function
- Domain: Set A (all inputs).
- Codomain: Set B (all possible outputs).
- Range: Set of actual outputs
{f(x) : x in A}.
Types of Functions
One-One (Injective) Function
Each element of A maps to a distinct element of B. f(x_1) = f(x_2) implies x_1 = x_2.
Many-One Function
Two or more elements of A map to the same element of B.
Onto (Surjective) Function
Every element of B has at least one pre-image in A. Range = Codomain.
Into Function
There exists at least one element in B with no pre-image in A.
Bijective Function
Both one-one and onto.
Real Functions
Functions whose domain and range are subsets of real numbers R.
Standard Real Functions
Identity Function: f(x) = x, Domain = R, Range = R.
Constant Function: f(x) = c, Domain = R, Range = {c}.
Polynomial Function: f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0, Domain = R.
Rational Function: f(x) = p(x)/q(x) where p(x), q(x) are polynomials. Domain = R - {x : q(x) = 0}.
Modulus Function: f(x) = |x| = {x if x >= 0, -x if x < 0}. Domain = R, Range = [0, infinity).
Signum Function: f(x) = {1 if x > 0, 0 if x = 0, -1 if x < 0}. Domain = R, Range = {-1, 0, 1}.
Greatest Integer Function: f(x) = [x] = greatest integer less than or equal to x. Domain = R, Range = Z.
Worked Examples
Example 1: Find the domain and range of f(x) = 1/sqrt(x - 3).
Solution: x - 3 > 0 implies x > 3. Domain = (3, infinity). Range = (0, infinity).
Example 2: If f(x) = x^2 + 3x + 2, find f(1), f(-2), and f(1/x).
Solution: f(1) = 1 + 3 + 2 = 6, f(-2) = 4 - 6 + 2 = 0, f(1/x) = 1/x^2 + 3/x + 2.
Common Mistakes
- Confusing range with codomain: Range is a subset of codomain, not necessarily equal.
- Vertical line test: A graph represents a function if no vertical line cuts it at more than one point.
- Horizontal line test for one-one: If a horizontal line cuts the graph at more than one point, the function is not one-one.
- Domain restrictions: Always exclude values making denominator zero or radicand negative for even roots.
ISC Exam Focus
- Theory: 70% weightage on definitions, types of functions, domain and range.
- Application: 30% on finding domain/range and identifying function types.
- Typical questions: finding domain and range of real functions, determining if a function is one-one/onto, graph sketching.
- 4-6 mark questions frequently appear from this chapter.
Self-Test Questions
Q1: Find the domain of f(x) = (x - 1)/(x^2 - 5x + 6).
Answer: x^2 - 5x + 6 = 0 gives x = 2, 3. Domain = R - {2, 3}.
Q2: Find the range of f(x) = |x - 3|.
Answer: Range = [0, infinity).
Q3: Is the function f : R -> R defined by f(x) = x^2 one-one?
Answer: No, because f(-1) = 1 = f(1) but -1 != 1. It is many-one.
Q4: If f(x) = 2x + 1 and g(x) = x^2, find f(g(2)) and g(f(3)).
Answer: g(2) = 4, f(4) = 9. f(3) = 7, g(7) = 49.
Q5: Find the domain and range of f(x) = 1/(x - 2).
Answer: Domain = R - {2}. Range = R - {0}.
Q6: Determine if f : Z -> Z defined by f(x) = x + 2 is bijective.
Answer: One-one: f(x_1) = f(x_2) => x_1 + 2 = x_2 + 2 => x_1 = x_2. Onto: for every y in Z, x = y - 2 in Z. Hence bijective.
