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: 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 |
