Unit 01 · Lesson 04 · Open-Age
Function Composition & Decomposition
Composition uses the output of one function as the input of another. The order matters, and the domain must satisfy both the inner function and the outer function after substitution.
Learning objectives
- Explain the central structure of function composition & decomposition using precise mathematical language.
- Apply the governing relationship and verify results with an alternate representation or substitution.
- State domain, unit, precision, and context restrictions whenever they affect the result.
- Analyze a misconception and communicate a correction with evidence.
Essential question: How do notation, transformations, composition, and inverse thinking reveal the structure and behavior of functions?
Governing relationship
(f∘g)(x)=f(g(x)).
Composition uses the output of one function as the input of another. The order matters, and the domain must satisfy both the inner function and the outer function after substitution.
Worked examples
Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.
Guided and independent practice
- Compute f(g(2)) for f(x)=1/x and g(x)=x−5.
- Find the domain of √(3−x²).
- Decompose 4(x−2)²+7 into simpler functions.
- Create one original example involving function composition & decomposition and solve it completely.
- Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
- Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.
Differentiated pathways
Foundation
Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.
Core
Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.
Extended
Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.
Application laboratory
Model a two-stage process such as discount then tax, conversion then fee, or speed then stopping distance.
Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.
Misconception and correction
Common misconception: Assuming composition is commutative. In general f∘g≠g∘f.
Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.
Mastery evidence
- Accurate symbolic work with preserved equivalence.
- Correct domain, interval, unit, and precision statements.
- At least one independent verification.
- A correction explaining why an incorrect approach fails.
- A clear application conclusion that distinguishes mathematical result from recommendation.
Reflection
What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?