What Is a Function?
A function f from set A to set B is a relation that assigns exactly one element of B to each element of A. We write f: A → B.
Key Terms:
Domain: Set A (all possible inputs)
Codomain: Set B (all possible outputs)
Range/Image: The set of actual outputs: {f(a) | a ∈ A}
Example: f: ℝ → ℝ, f(x) = x²
Domain: ℝ, Codomain: ℝ, Range: [0, ∞)Function Notation
f: A → B means for every a ∈ A, there is exactly one f(a) ∈ B
Examples:
f(x) = 2x + 1 Linear function
g(x) = x² Quadratic function
h(x) = |x| Absolute value function
Not a function: x² + y² = 1 (y is not uniquely determined by x)Special Functions
Identity: id_A(x) = x for all x ∈ A
Constant: f(x) = c for all x ∈ A
Floor: ⌊x⌋ = greatest integer ≤ x
Ceiling: ⌈x⌉ = smallest integer ≥ x
Modular: f(a) = a mod n