Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 20: Special Functions

Lesson 5: Piecewise Defined Functions

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how piecewise defined functions apply different rules to different parts of a function's domain. Using bracket notation, students practice evaluating outputs, solving equations like f(x) = x, determining whether a piecewise function has an inverse, and graphing each piece to analyze continuity. The lesson also covers rewriting absolute value expressions such as f(x) = |3 − 7x| as piecewise defined functions without absolute value signs.

Section 1

Writing Piecewise Functions with Multiple Conditions

Property

A piecewise function is defined using multiple rules for different input ranges:

f(x)={rule1if condition1rule2if condition2rule3if condition3⋮⋮f(x) = \begin{cases} \text{rule}_1 & \text{if condition}_1 \\ \text{rule}_2 & \text{if condition}_2 \\ \text{rule}_3 & \text{if condition}_3 \\ \vdots & \vdots \end{cases}

Examples

Section 2

Converting Absolute Value Expressions to Piecewise Functions

Property

To convert an expression containing absolute values to piecewise form: find the zeros of each expression inside absolute value bars, use these zeros to divide the domain into intervals, then determine the sign of each absolute value expression on each interval. For ∣g(x)∣|g(x)|: if g(x)≥0g(x) \geq 0 then ∣g(x)∣=g(x)|g(x)| = g(x), if g(x)<0g(x) < 0 then ∣g(x)∣=−g(x)|g(x)| = -g(x).

Examples

Section 3

Fixed Points of Piecewise Functions

Property

A fixed point of a function occurs where f(x)=xf(x) = x. For piecewise functions, solve f(x)=xf(x) = x separately within each piece's domain, then verify that solutions satisfy the domain conditions for that piece.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Writing Piecewise Functions with Multiple Conditions

Property

A piecewise function is defined using multiple rules for different input ranges:

f(x)={rule1if condition1rule2if condition2rule3if condition3⋮⋮f(x) = \begin{cases} \text{rule}_1 & \text{if condition}_1 \\ \text{rule}_2 & \text{if condition}_2 \\ \text{rule}_3 & \text{if condition}_3 \\ \vdots & \vdots \end{cases}

Examples

Section 2

Converting Absolute Value Expressions to Piecewise Functions

Property

To convert an expression containing absolute values to piecewise form: find the zeros of each expression inside absolute value bars, use these zeros to divide the domain into intervals, then determine the sign of each absolute value expression on each interval. For ∣g(x)∣|g(x)|: if g(x)≥0g(x) \geq 0 then ∣g(x)∣=g(x)|g(x)| = g(x), if g(x)<0g(x) < 0 then ∣g(x)∣=−g(x)|g(x)| = -g(x).

Examples

Section 3

Fixed Points of Piecewise Functions

Property

A fixed point of a function occurs where f(x)=xf(x) = x. For piecewise functions, solve f(x)=xf(x) = x separately within each piece's domain, then verify that solutions satisfy the domain conditions for that piece.

Examples