Definitions [3]
Definition: Function
f: X → Y is a function if each element of X is associated with a unique element of Y
Definition: Domain & Codomain
- Domain (X): Set of all input values
- Codomain (Y): Set of all possible outputs
Definition: Range
- Range: Set of actual output values of f
- Range ⊆ Codomain
Key Points
Key Points: Types of Functions
| Type of Function | Condition | Key Idea |
|---|---|---|
| One-One (Injective) | f(x₁) = f(x₂) ⇒ x₁ = x₂ | Different inputs → different outputs |
| Onto (Surjective) | Range = Codomain | Every element of the codomain is mapped |
| Into Function | Range ⊂ Codomain | Some elements of the codomain are not mapped |
| Many-One Function | x₁ ≠ x₂ but f(x₁) = f(x₂) | Different inputs → same output |
| Bijective Function | One-one + Onto | Perfect mapping (1-1 and onto) |
Key Points: Algebra of Functions
| Operation | Result | Domain |
|---|---|---|
| Addition | (f + g)(x) = f(x) + g(x), ∀ x ∈ D₁ ∩ D₂ | D₁ ∩ D₂ |
| Subtraction | (f − g)(x) = f(x) − g(x), ∀ x ∈ D₁ ∩ D₂ | D₁ ∩ D₂ |
| Multiplication | (fg)(x) = f(x) · g(x), ∀ x ∈ D₁ ∩ D₂ | D₁ ∩ D₂ |
| Quotient | \[\frac{f}{g}(x)=\frac{f(x)}{g(x)}\], ∀ x ∈ D₁ ∩ D₂ (g(x) ≠ 0) | D₁ ∩ D₂ − {x : g(x) = 0} |
| Multiplication by a scalar | (cf)(x) = cf(x), ∀ x ∈ D₁ | D₁ |
