SYSTEMATIC MATHEMATICS

Algebra & Functions

This course contains 36 visible knowledge chapters across eight content-sized units of lengths 4, 4, 5, 4, 4, 5, 5, and 5. Begin by reading letters, parentheses, fraction bars, and exponents as an operation tree, then translate ordinary language without reversing the relationship. Build equations and inequalities as solution-set statements; classify identities, contradictions, lines, half-planes, functions, piecewise rules, systems, polynomials, finite rational-root candidates, rational-expression arithmetic, radicals, exponentials, logarithms, sequences, finite sums, and proportional-variation models. Linear objectives are solved only after feasible regions are built, and rational operations are taught before rational functions use them. Every transformation names its domain, reversible step, complete example, independent check, and first failure boundary.

Before this course: Mathematical Foundations: exact arithmetic with signed numbers, fractions, ratios, powers, roots, units, estimation, and divisibility.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A variable ranges over a stated domain

Objective: Why is “x is a number” insufficient information when solving √(x−1)=x−3?

A variable is a symbol whose value may vary within a specified set. The symbol alone does not tell us whether its values are integers, real numbers, positive measurements, or members of some other set. That domain is part of the mathematical statement. In the expression 1/x, real-number algebra excludes x=0 because division by zero is undefined. In √x, a real-valued interpretation requires x≥0. Ignoring these conditions changes the object being studied.

A parameter also uses a symbol, but its role is different: it is held fixed while other variables change. In y=mx+b, x is usually the input, y the output, and m and b parameters selecting one line from a family. Roles can change when the question changes, so a careful solution names them. Units and context can narrow a domain further: an algebraic solution t=−3 may be a real number yet fail as elapsed time after an experiment begins.

For real-valued expressions, a denominator must be nonzero and an even-indexed radical must have a nonnegative radicand.

Division a/b is defined as the unique number q satisfying bq=a. If b=0, then 0q is always 0, so it cannot equal nonzero a and cannot determine a unique q when a=0.

For a real even root r=√[2k](u), the defining relation is r^(2k)=u with r≥0. Every real even power is nonnegative, so negative u cannot satisfy that relation.

Intersecting all required conditions gives the actual domain. Contextual conditions such as length>0 are then intersected with the algebraic domain rather than substituted informally at the end.

Find the real domain of F(x)=√(5−2x)/(x−3), and explain each boundary.

  1. The square root requires 5−2x≥0. Subtract 5 and divide by −2, reversing the inequality, to obtain x≤5/2.
  2. The denominator requires x−3≠0, so x≠3. However, 3 is already outside x≤5/2.
  3. Intersect the conditions. Every x≤5/2 gives a nonnegative radicand and a nonzero denominator, including x=5/2 where the numerator is zero.

Result: The domain is (−∞,5/2].