- §1.1MappingsEvery element of A fires exactly one arrow. That is a mapping.
- §1.2FunctionsA function maps numbers to numbers; to move its graph, move every point.
- §2.1Odd and Even FunctionsEven folds onto itself, odd survives a half-turn; any f is even + odd.
- §2.2MonotonicityIncreasing means the right point is always higher; prove it via f(x₂) − f(x₁).
- §2.3PeriodicityShift the graph by T onto itself and T is a period; two mirrors make one too.
- §3.1Quadratic FunctionsDiscriminant, axis and end values together lock both roots in an interval.
- §3.2Power, Exponential and Logarithmic Functionsaˣ and logₐx mirror each other in y = x, but they need not meet on that mirror.
- §3.3The Function f(x) = x + a²/xTwo terms with fixed product a² have least sum 2a, exactly when they are equal.
- Ch. 4Maxima and Minima of FunctionsA max or min is a bound plus a point that reaches it. The methods find bounds.
- Ch. 5Solving Problems by Constructing a FunctionRead an expression as a value of a function; let its properties do the work.
- §6.1Iteration of a FunctionFeed a function's output back in, again and again. A cobweb shows the orbit.
- §6.2Finding the n-th IterateGuess and prove; use a fixed point; or build a bridge from f to a simpler g.
- §6.3Applications of IterationRepeat one map and ask where the orbit goes. Fixed points, shrinking, cycles.
- §7.1Functional Equations: SubstitutionRewrite the equation with x replaced until f(x) returns, then eliminate.
- §7.2Functional Equations: Plugging in ValuesIt holds for all x and y, so plug in the values that tell you most about f.
- §7.3Functional Equations: Cauchy's MethodSolve on ℕ, climb to ℤ and ℚ, then reach ℝ with monotonicity or continuity.
- §7.4Functional Equations: RecursionOn the positive integers, a functional equation is just a recurrence. Solve it.