A function f: A -> B assigns to each element of A (domain) exactly one element of B (codomain). The range f(A) is the set of all output values, always a subset of the codomain. Injective (one-to-one): f(a1) = f(a2) implies a1 = a2 — no two inputs share an output. Surjective (onto): range equals codomain — every element of B is hit. Bijective: both injective and surjective — establishes a perfect pairing. Tests: for injectivity, show the function is strictly monotonic (for real functions) or use the definition directly. For surjectivity, solve y = f(x) for x and check that a solution exists for every y in B. For bijectivity, verify both. Only bijections have inverse functions. The horizontal line test on graphs determines injectivity (each horizontal line intersects the graph at most once).