HSSLIVE Plus One Maths Chapter 2: Relations and Functions Notes

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Relations and functions are powerful tools that describe connections between mathematical objects. This chapter guides students through the formal definitions of relations, types of relations, and the special properties that define functions. Students will explore domains, ranges, composition of functions, and inverse functions while developing the ability to represent these concepts algebraically, graphically, and through mapping diagrams. These concepts are crucial for modeling real-world relationships in fields ranging from economics to physics.

Chapter 2: Relations and Functions

Relations:

A relation is a connection between two sets. Mathematically, a relation R from set A to set B is a subset of the Cartesian product A × B.

Types of Relations:

  • Reflexive: For all a ∈ A, (a, a) ∈ R
  • Symmetric: If (a, b) ∈ R, then (b, a) ∈ R
  • Transitive: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
  • Equivalence Relation: Relation that is reflexive, symmetric, and transitive

Functions:

A function f from set A to set B is a relation where each element of A is paired with exactly one element in B.

Representation: f: A → B

Domain: Set A (input values) Co-domain: Set B (possible output values) Range: Actual output values obtained when applying f to all elements in A

Types of Functions:

  1. One-to-One (Injective): Different elements in domain have different images.
    • For all a, b ∈ A, if f(a) = f(b), then a = b
  2. Onto (Surjective): Every element in the co-domain has at least one pre-image.
    • For all b ∈ B, there exists a ∈ A such that f(a) = b
  3. Bijective: Both one-to-one and onto.

Operations on Functions:

  1. Addition: (f + g)(x) = f(x) + g(x)
  2. Subtraction: (f – g)(x) = f(x) – g(x)
  3. Multiplication: (f · g)(x) = f(x) · g(x)
  4. Division: (f/g)(x) = f(x)/g(x), where g(x) ≠ 0
  5. Composition: (f ∘ g)(x) = f(g(x))

Inverse Functions:

If f: A → B is bijective, then its inverse function f⁻¹: B → A exists such that:

  • f⁻¹(f(a)) = a for all a ∈ A
  • f(f⁻¹(b)) = b for all b ∈ B

 

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