BetterGrades Precalculus · Unit 2 · Lesson

Piecewise-defined functions

Evaluate, graph, and interpret functions that use different rules on different parts of the domain.

Opening

Start with the situation

A piecewise function uses different rules on nonoverlapping parts of one domain.

The same dependency may arrive as a story, table, graph, or formula. Learning to preserve the inputs, outputs, units, and domain while moving among those forms is the central language of the course.

Before you begin

Prerequisite check

Use function notation from the algebra and function readiness unit.
Read ordered pairs and interval notation.
Attach units to contextual quantities.

Core explanation

Explanation

Locate the input interval first, evaluate with that rule, and graph each restricted piece with correct open or closed endpoints.

At a boundary, continuity requires both one-sided behaviors and the defined value to agree.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through piecewise rule selector, endpoint gallery, or another equivalent representation.

Conceptual reading

What the idea is really doing

A function is a dependency, not merely an equation. Inputs, outputs, units, and domain must agree in words, tables, graphs, and formulas; each representation should tell the same mathematical story.

This lesson narrows that lens to one goal: evaluate, graph, and interpret functions that use different rules on different parts of the domain. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Locate the input interval first.
Evaluate with that rule.
Graph each restricted piece with correct open or closed endpoints.

Verification: Choose two representations and make them verify one another. A table can test a formula, a graph can expose a domain or range claim, and units can reveal a model that is algebraically neat but conceptually wrong.

Foundation walkthrough

Plan before calculating

Problem

$f=x+2$ for $x<1$ and $x^2$ for $x\ge 1$ ; find $f(-3),f(1),f(2)$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Locate the input interval first, evaluate with that rule, and graph each restricted piece with correct open or closed endpoints.
Conclusion: $-1,1,4$
Why the check works: The boundary belongs to the second rule.

Worked examples

See the idea in three forms

foundation example

$f=x+2$ for $x<1$ and $x^2$ for $x\ge 1$ ; find $f(-3),f(1),f(2)$ .

Solution $-1,1,4$

The boundary belongs to the second rule.

representation example

Meaning of overlapping intervals.

SolutionThe definition may assign two outputs.

This example expresses piecewise-defined functions in a second form.

transfer example

Meaning of omitted boundary.

SolutionThe function is undefined there.

At a boundary, continuity requires both one-sided behaviors and the defined value to agree.

Piecewise rule selector. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The boundary belongs to the second rule. — **Anchor figure · Piecewise rule selector**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The boundary belongs to the second rule.

Endpoint gallery. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for piecewise-defined functions. — **Mechanism figure · Endpoint gallery**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for piecewise-defined functions.

Boundary behavior comparison. Compare the valid path with the tempting shortcut. The figure shows why evaluating every formula or including both boundary points leads to a false conclusion. — **Comparison and error figure · Boundary behavior comparison**
Compare the valid path with the tempting shortcut. The figure shows why evaluating every formula or including both boundary points leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is evaluating every formula or including both boundary points.

Check yourself

Write |x| piecewise.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Write |x| piecewise.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Meaning of overlapping intervals.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Meaning of omitted boundary.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Line style for excluded endpoint.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: $-1,1,4$ . Use the foundation problem as evidence: $f=x+2$ for $x<1$ and $x^2$ for $x\ge 1$ ; find $f(-3),f(1),f(2)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of piecewise-defined functions that it illustrates: Meaning of overlapping intervals.

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is evaluating every formula or including both boundary points.

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: $f=x+2$ for $x<1$ and $x^2$ for $x\ge 1$ ; find $f(-3),f(1),f(2)$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Meaning of omitted boundary. Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for piecewise-defined functions, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Average rate of change, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Lippman and Rasmussen, Precalculus Volume 1
Stitz and Zeager, Precalculus

No long source passage is reproduced.

f=x+2f=x+2f=x+2 for x<1x<1x<1 and x2x^2x2 for x≥1x\ge 1x≥1; find f(−3),f(1),f(2)f(-3),f(1),f(2)f(−3),f(1),f(2).

f=x+2f=x+2f=x+2 for x<1x<1x<1 and x2x^2x2 for x≥1x\ge 1x≥1; find f(−3),f(1),f(2)f(-3),f(1),f(2)f(−3),f(1),f(2).

Meaning of overlapping intervals.

Meaning of omitted boundary.

Original BetterGrades manuscript, rights-separated references.

$f=x+2$ for $x<1$ and $x^2$ for $x\ge 1$ ; find $f(-3),f(1),f(2)$ .

$f=x+2$ for $x<1$ and $x^2$ for $x\ge 1$ ; find $f(-3),f(1),f(2)$ .