Sets, Relations and Functions

A function f: A → B assigns each element of A (domain) to exactly one element of B (codomain). The set of values f actually takes is the range ⊆ B.

One-one (injective): different inputs give different outputs.
Test: f(x₁) = f(x₂) ⇒ x₁ = x₂. Or graphically: every horizontal line crosses the graph at most once.
Example: f(x) = 2x is one-one. f(x) = x² on ℝ is not (since f(2) = f(−2) = 4).

Onto (surjective): every element of the codomain is hit.
Test: range = codomain.
Example: f: ℝ → ℝ, f(x) = x³ is onto. f: ℝ → ℝ, f(x) = x² is not onto (negative numbers aren't in the range), but f: ℝ → [0, ∞) is.

Bijection = one-one AND onto. Only bijections have inverses.

Finding f⁻¹:

1. Write y = f(x).
2. Solve for x in terms of y.
3. Swap variables: x = f⁻¹(y), then rename y to x.

Example: f(x) = 3x + 5. Then y = 3x + 5 → x = (y − 5)/3. So f⁻¹(x) = (x − 5)/3.

Composition: (g ∘ f)(x) = g(f(x)). Generally NOT commutative: g ∘ f ≠ f ∘ g.

Key identity for inverses: (f⁻¹ ∘ f)(x) = x = (f ∘ f⁻¹)(x).