Chapter 2 — Functions and Dynamic Relations¶

If the real numbers are the raw material of mathematical analysis, functions are its dynamics: the rules that govern the dependence, transformation and variation of magnitudes.

In this chapter we move from the intuitive idea of a table of values to the formal definition of a function as a set of pairs, and we build the tools to study how functions behave, how they combine and how they are transformed.

Contents of the chapter¶

2.1 — The Notion of Function: Cartesian product, relations, definition of function by existence and uniqueness, domain, codomain and image, and representations of a function.
2.2 — Explicit and Implicit Domain: the implicit-domain convention, the structural restrictions of the available operations, restriction and equality of functions.
2.3 — Visualization and Basic Topology: the vertical-line test, domain and image read from the graph, zeros and sign, even and odd functions, and monotonicity.
2.4 — Catalog of Prototype Functions: the six prototypes \(x\), \(x^2\), \(x^3\), \(|x|\), \(\sqrt{x}\) and \(\frac{1}{x}\), unbounded growth, and vertical and horizontal asymptotes.
2.5 — Geometry of the Space of Functions: translations, scalings, reflections and parity, and the order of execution in composite transformations.
2.6 — Algebra and Composition: arithmetic operations between functions and their domain, the composition \(g \circ f\), associativity and lack of commutativity, and decomposition into elementary functions.
2.7 — Injectivity and Inverse Function: the horizontal-line test, surjectivity and bijectivity, construction of \(f^{-1}\), restricting the domain to invert, and symmetry about the line \(y = x\).
2.8 — Optimization and Extreme Values: absolute and relative extrema, the vertex of the parabola, extrema through monotonicity, and the inequality between the arithmetic and geometric means.
2.9 — Closing: From Quantity to Change: historical and epistemological essay on the evolution of the concept of function, from Babylonian tables to the set-theoretic definition.
2.10 — Closing: Synthesis and Checklist: map of the chapter's dependencies and a list of competencies by section.
2.11 — Closing: Review Problems: problems at three levels, with an answer key.
2.12 — Closing: Model Exam: exam with a grading guide.

"A function is not just a formula; it is the law of a transformation."