Learn on PengienVision, Algebra 2Chapter 1: Linear Functions and Systems

Lesson 3: Piecewise-Defined Functions

In this Grade 11 enVision Algebra 2 lesson from Chapter 1, students learn to graph and interpret piecewise-defined functions, including step functions, where different rules apply over different parts of the domain. Students practice writing piecewise notation, identifying domain and range, and determining intervals where the function is increasing or decreasing. Real-world contexts like hourly wages with overtime help students understand why a single linear rule sometimes cannot model an entire situation.

Section 1

Piecewise Function Definition and Notation

Property

A piecewise-defined function is a function defined by different expressions over different intervals of its domain. The general notation is:

f(x)={expression1if condition1expression2if condition2⋮⋮expressionnif conditionnf(x) = \begin{cases} \text{expression}_1 & \text{if condition}_1 \\ \text{expression}_2 & \text{if condition}_2 \\ \vdots & \vdots \\ \text{expression}_n & \text{if condition}_n \end{cases}

Section 2

Graphing Piecewise Functions with Boundary Points

Property

When graphing piecewise functions, use closed circles (•) to indicate points included in the domain of a piece, and open circles (○) to indicate points not included. At boundary points where domain intervals meet, only one piece can contain the boundary value.

Examples

Section 3

Writing Piecewise Functions from Graphs

Property

To write a piecewise function from a graph: identify each piece's domain interval, determine the equation for each piece using slope-intercept form y=mx+by = mx + b, and note whether boundary points use closed circles (included with ≤\leq or ≥\geq) or open circles (excluded with << or >>).

Examples

Section 4

Absolute Value as Piecewise Function

Property

The absolute value function can be written as a piecewise-defined function with two pieces:

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Piecewise Function Definition and Notation

Property

A piecewise-defined function is a function defined by different expressions over different intervals of its domain. The general notation is:

f(x)={expression1if condition1expression2if condition2⋮⋮expressionnif conditionnf(x) = \begin{cases} \text{expression}_1 & \text{if condition}_1 \\ \text{expression}_2 & \text{if condition}_2 \\ \vdots & \vdots \\ \text{expression}_n & \text{if condition}_n \end{cases}

Section 2

Graphing Piecewise Functions with Boundary Points

Property

When graphing piecewise functions, use closed circles (•) to indicate points included in the domain of a piece, and open circles (○) to indicate points not included. At boundary points where domain intervals meet, only one piece can contain the boundary value.

Examples

Section 3

Writing Piecewise Functions from Graphs

Property

To write a piecewise function from a graph: identify each piece's domain interval, determine the equation for each piece using slope-intercept form y=mx+by = mx + b, and note whether boundary points use closed circles (included with ≤\leq or ≥\geq) or open circles (excluded with << or >>).

Examples

Section 4

Absolute Value as Piecewise Function

Property

The absolute value function can be written as a piecewise-defined function with two pieces:

∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Examples