≥Better Grades
Menu

BetterGrades Algebra · Unit A12 · Unit Hub

Unit A12: Functions as a Unifying Language

Consolidate earlier relationships into function language without preempting full Precalculus theory. How can one input-output framework organize the algebraic families already studied?

Act XII - Organizing algebra around input and output

The 8-lesson path

Relations become functions when each input has one output. Function notation separates evaluation from solving; domain and range collect restrictions and outputs; piecewise functions and arithmetic unify earlier examples.

Governing question

How can one input-output framework organize the algebraic families already studied?

Consolidate earlier relationships into function language without preempting full Precalculus theory.

Mastery design

Know when to continue

Students move among table, graph, formula, and context and distinguish evaluation from solving.

Unit outcomes
  1. Determine whether a relation is a function.
  2. Evaluate functions and solve output equations.
  3. Determine domain and range from formulas, graphs, and contexts.
  4. Interpret graph behavior and piecewise rules.
  5. Combine functions arithmetically while tracking domain.
Core sequence

Lessons

Move in order when this material is new; use prerequisites and repair links when entering mid-unit.

  1. 01Relations and functionsDefine a function as exactly one output for each allowed input.
  2. 02Function notation and evaluationInterpret f(a)f(a) as the output corresponding to input a rather than multiplication.
  3. 03Solving f(x)=kFind every input that produces a requested output.
  4. 04DomainDetermine allowed inputs from context, denominators, and even roots.
  5. 05Range and graph behaviorRead output sets, intervals of change, extrema, and intercepts from graphs and contexts.
  6. 06Piecewise-defined functionsApply different rules on different input intervals and manage endpoints.
  7. 07Arithmetic with functionsAdd, subtract, multiply, and divide function outputs while tracking the combined domain.
  8. 08Comparing algebraic familiesIdentify change patterns, restrictions, and characteristic graph shapes across major families.

Practice around the path

Review, mixed practice, mastery, and investigation

Every route contains a fixed set of concrete questions, with detailed solutions available after an attempt.

ReviewCumulative reviewRevisit this unit with prior-unit retrieval.PracticeMixed practiceClassify the structure before choosing a method.MasteryMastery checkUse the supplied grading and cumulative-share contract.Response guideMastery response guideAttempt-gated rubric support; no invented canonical answers.

Source & rights

Original storyboard, rights-separated references.

Public page content comes from the BetterGrades Algebra editorial storyboard supplied by the owner. Reference books named in provenance remain separate and are not copied into the application.