Chapter 2 — Functions and Dynamic Relations¶
2.10 — Mathematical Synthesis and Competency Checklist¶
Synthesis: the backbone of the chapter¶
The eight sections of the chapter are not eight independent topics, but eight successive views of a single decision made in §2.1: to require that a relation satisfy existence and uniqueness in order to be called a function. Before any checklist, it is worth tracing how that decision runs through the whole chapter.
Uniqueness, formulated in §2.1 as a condition on sets, reappears twice in geometric form, with opposite roles. In §2.3 it is the vertical-line test, which no function can violate because it is its definition seen in the plane. In §2.7 it is the horizontal-line test, which expresses an additional condition, injectivity, that a function may or may not satisfy, and on which the existence of \(f^{-1}\) depends. Between the two appearances lies the whole difference between a property that defines and a property that one has or does not have; confusing them is the most frequent conceptual error of the chapter.
The domain follows a parallel path. It is part of the function from §2.1 on, it is computed in §2.2 from the restrictions of the available operations, it is shifted or rescaled predictably by the horizontal transformations of §2.5 and left unchanged by the vertical ones, it is obtained by intersection in the algebra of functions of §2.6, and it is exchanged with the image when \(f^{-1}\) is built in §2.7. The distinction between codomain and image, which in §2.1 looks like a technicality, is exactly what decides in §2.7 whether a function is surjective.
The catalog of §2.4 and the transformations of §2.5 work as a pair: the first provides six objects analyzed in detail; the second, a way of generating many new functions from them without repeating the analysis. Section 2.6 adds a second means of generation, composition, and shows that the transformations of §2.5 are a particular case of it: composing with functions of the form \(mx + n\). Section 2.7 inverts functions, sometimes after restricting their domain. Section 2.8 returns to earlier tools with a new purpose: the vertex of the parabola is the completing of the square of §1.3, and the notion of relative extremum uses the same neighborhood of §1.1 that §2.4 had used for asymptotic behavior. No technical instrument of the chapter was introduced twice: each one was built only once and used in every context where it was needed again.
| Section | Central object | Relies on | Reappears in |
|---|---|---|---|
| 2.1 | Function as a relation with existence and uniqueness; codomain and image | \(A\times B\) and ordered pairs (§1.8); \(\subseteq\) | Vertical line (2.3); surjectivity (2.7); existence and uniqueness of \(f^{-1}\) (2.7) |
| 2.2 | Implicit domain; restriction and equality of functions | Inverses, roots and exponents (§1.1, §1.2); simplification (§1.4) | Domain of operations and of composition (2.6); restricting to invert (2.7) |
| 2.3 | Zeros, sign, parity, monotonicity | Definition of function (2.1); sign analysis (§1.7) | Prototypes (2.4); parity and reflection (2.5); injectivity (2.7); extrema (2.8) |
| 2.4 | Six prototypes; asymptotes and limit | Domain (2.2); properties (2.3); neighborhood (§1.1) | Transformations (2.5); composition (2.6); inverse of \(x^2\) (2.7) |
| 2.5 | Translation, scaling and reflection of graphs | Prototypes (2.4); solving equations (§1.5) | Composition with linear functions (2.6); extrema (2.8) |
| 2.6 | Operations on functions; composition \(g\circ f\) | Domain (2.2); prototypes (2.4) | Characterization of the inverse (2.7) |
| 2.7 | Injectivity, surjectivity, \(f^{-1}\) | Codomain and image (2.1); monotonicity (2.3); composition (2.6) | — |
| 2.8 | Absolute and relative extrema | Completing the square (§1.3); monotonicity (2.3); neighborhood (§1.1) | — |
Competency checklist¶
The following list is not an exam but a self-assessment tool to use before one. It is organized in the three levels the book distinguishes: procedural, carrying out a known technique; conceptual, explaining why it works or recognizing it in an unusual form; and proof-based, producing or reconstructing the argument that justifies it. Whoever masters only the first level can solve routine exercises, but is not yet ready to take on the next chapter with the solidity it demands.
§2.1 — The function as a relation
- [ ] Procedural: given a set of ordered pairs, decide whether it defines a function from a given set to another.
- [ ] Conceptual: explain what fails, existence or uniqueness, in each of two examples of relations that are not functions.
- [ ] Proof-based: given a formula and two sets, justify with the definition, and not with a graph, whether it defines a function between them, and compute its image by proving both inclusions.
§2.2 — Implicit domain
- [ ] Procedural: compute the implicit domain of a function combining quotients, even roots and fractional exponents.
- [ ] Conceptual: explain why two formulas that agree after simplification can define different functions.
- [ ] Proof-based: decide and justify whether two functions given by different formulas are equal according to the definition of §2.2.
§2.3 — Global behavior
- [ ] Procedural: determine zeros, sign, parity and monotonicity of a function given by a quotient of polynomials.
- [ ] Conceptual: explain why the vertical-line test is the definition of function and not a separate rule, and why the sign may change at a point excluded from the domain.
- [ ] Proof-based: prove, with the definition of monotonicity and without the graph, that a given function is increasing or decreasing on an interval.
§2.4 — Prototypes and asymptotic behavior
- [ ] Procedural: recall domain, image, parity and monotonicity of the six prototypes.
- [ ] Conceptual: distinguish between growing without bound near a point, which gives a vertical asymptote, and approaching a value as the variable grows without bound, which gives a horizontal asymptote.
- [ ] Proof-based: given \(M > 0\), or \(\epsilon > 0\), find the \(\delta\), or the \(N\), that proves the asymptotic behavior of a function such as \(\frac{1}{x}\).
§2.5 — Transformations of graphs
- [ ] Procedural: graph a function obtained from a prototype by translation, scaling or reflection, identifying \(a\), \(h\), \(c\) and \(k\).
- [ ] Conceptual: explain why \(f(x - h)\) shifts the graph to the right when \(h > 0\).
- [ ] Proof-based: prove, using the correspondence of points, the rule for a transformation that does not appear in the book.
§2.6 — Algebra and composition
- [ ] Procedural: compute \((f + g)(x)\), \((f \cdot g)(x)\) and \((g \circ f)(x)\) with their domains.
- [ ] Conceptual: explain why the domain of \(g \circ f\) requires a condition on \(f(x)\) and not only on \(x\).
- [ ] Proof-based: decompose a function into elementary pieces, justify the order chosen and check that the domain of the composition agrees with that of the function.
§2.7 — Injectivity and inverse function
- [ ] Procedural: decide whether a function is injective and, if so, find \(f^{-1}\), first restricting the domain if needed.
- [ ] Conceptual: explain why injectivity and surjectivity are both needed for \(f^{-1}\) to be a function.
- [ ] Proof-based: verify that a proposed formula is the inverse of \(f\) by checking \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(y)) = y\).
§2.8 — Extrema
- [ ] Procedural: find the vertex of a quadratic function and classify it as a maximum or a minimum.
- [ ] Conceptual: explain why every absolute extremum is relative, why the converse is false, and why the extremum of a formula may fail to solve an applied problem.
- [ ] Proof-based: prove the AM-GM inequality and apply it to a constrained optimization problem not solved in the book.
With this synthesis and checklist, the reader can self-assess before the review problems and the chapter exam, organized according to these same three levels.