Unit 01 · Lesson 06 · Open-Age

Piecewise, Absolute-Value & Step Functions

Piecewise functions use different rules on different parts of the domain. Endpoint symbols and context determine whether a boundary value is included, excluded, or shared.

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Learning objectives

Explain the central structure of piecewise, absolute-value & step functions 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(x)={rule₁ for condition₁; rule₂ for condition₂; …}.

Piecewise functions use different rules on different parts of the domain. Endpoint symbols and context determine whether a boundary value is included, excluded, or shared.

Worked examples

Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.

Evaluate f(2) if f(x)=x+1 for x<2 and x² for x≥2.Because 2 belongs to x≥2, f(2)=4.

Model a parking fee: 5 for the first hour, then 3 per additional hour.One model is C(t)=5 for 0<t≤1 and C(t)=5+3(t−1) for t>1.

Guided and independent practice

Evaluate a three-rule piecewise function at boundary inputs.
Graph |x−3| as a piecewise linear function.
Design a tiered pricing model.
Create one original example involving piecewise, absolute-value & step functions 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

Represent a real policy, tariff, tax bracket, or service fee with a piecewise function.

Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.

Misconception and correction

Common misconception: Using the wrong rule at an endpoint because < and ≤ are treated as interchangeable.

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?