Unit 01 · Lesson 03 · Week 1 · Calculus Readiness & Function Fluency
Transformations, Composition & Inverse Structure
Coordinate algebra, function notation, units, and multiple representations establish the language used to describe change. A defensible solution identifies the function, the input and output quantities, the relevant domain, and the meaning of the result. This lesson emphasizes transformations, composition & inverse structure through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in transformations, composition & inverse structure.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do function structure, representations, units, and algebraic fluency prepare us to analyze change and accumulation?
Visual model
Coordinate algebra, function notation, units, and multiple representations establish the language used to describe change. A defensible solution identifies the function, the input and output quantities, the relevant domain, and the meaning of the result.
\frac{f(b)-f(a)}{b-a}(f\circ g)(x)=f(g(x))f^{-1}(f(x))=x
Worked reasoning
Misconception check: A calculator display is evidence, not an explanation. Domain, units, and reasonableness still must be justified.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Transformations, Composition & Inverse Structure
Student Worksheet · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Evaluate f(3) for f(x)=2x²−3x+1.
State the domain of g(x)=√(x−5).
For h(x)=1/(x+2), state the excluded input.
If f(x)=2x+3 and g(x)=x², find f(g(2)).
Convert 150° to radians.
Explain why units belong in a rate-of-change statement.
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.