The Function Machine

Unit 5 installs the course's central object — the function, a machine that gives each input exactly one output — and everything after is machinery: intake rules, gauges, wiring, editing, and reverse gear.

One rule creates the object: each input, exactly one output. Everything else in the unit is a consequence dressed as a topic. Notation names the machine and the crank — f(4) = 11 is a receipt, not a multiplication. The domain is the intake manifold, and the domain hunt is hazard removal: zero denominators out, negative even radicands out, survivors written as intervals. The range is what the machine actually ships, never what it might. The gauges come next: average rate of change reads slope on any function — 120 miles over 2 hours is 60 either way you dress it — and increasing, decreasing, and the local extremes describe the ride left to right. Then the machinery compounds. The algebra of functions merges machines pointwise; composition wires them in series, and the wiring is directional: f(g(2)) = 9 while g(f(2)) = 5, with the product 12 a third thing entirely. Transformations edit a machine without rebuilding it — outside edits move outputs, inside edits move inputs and run backward, which is why (x − 1) slides right and f(2x) narrows. And a one-to-one machine earns reverse gear: the inverse trades every (4, 11) for (11, 4), certified by composition running home to x, pictured as a reflection across y = x. The unit's twins both misread the machine. Assumed linearity splits f(1 + 4) into 17 when the machine says 25 — the missing-middle dream reborn. The reciprocal inverse reads f⁻¹ as 1/f and answers 1/25 where the reverse machine says 4 — an honest exponent rule loitering where there is no exponent. The cure for both is the same discipline: the notation means what it means, not what it resembles.