Precalculus — Functions
Functions as machines: notation, composition, transformations, inverses, and rational functions with asymptotes.
Calculus studies how functions change — so first, get fluent in functions themselves. A function is a machine: feed it a number, it returns exactly one number. Here you can build real ones and feed them live.
- 01 Function notation f(x) = x² − 3x defines the machine; f(4) runs it. The definition below is a real function — every cell after it can call it.
- 02 Composition — machines feeding machines f(g(x)) wires g's output into f's input. Order matters — composition is not commutative, and the checks prove it with a concrete x:
- 03 Transformations Every graph manipulation is arithmetic on the formula: f(x) + k slides UP by k f(x − h) slides RIGHT by h (the minus sign fools everyone once) a·f(x) stretches…
- 04 Inverse functions An inverse undoes: if f sends 4 → 4, then f⁻¹ sends 4 → 4 back. Only one-to-one functions have inverses — that's why we restrict x ≥ 0 before inverting x²:
- 05 Rational functions & asymptotes Divide polynomials and the graph grows asymptotes — lines the curve approaches but never touches. (x² − 1)/(x − 2) blows up vertically at x = 2, where the…
next course: Calculus 1 — Limits & Derivatives →