Learn on PengienVision, Algebra 1Chapter 5: Piecewise Functions

Lesson 2: Piecewise-Defined Functions

In this Grade 11 enVision Algebra 1 lesson, students learn to graph and apply piecewise-defined functions — functions that use different rules for different intervals of the domain. The lesson covers how to express absolute value functions in piecewise notation, identify increasing and decreasing intervals from a graph, and interpret real-world piecewise models such as tiered utility billing and variable pricing structures.

Section 1

Piecewise Function Definition and Cases Notation

Property

A piecewise-defined function is a function that uses different rules (expressions) for different intervals of its domain. The standard notation uses cases:

f(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

Interval Notation

Property

An interval is a set that consists of all the real numbers between two numbers $a$ and $b$ .

The closed interval $[a, b]$ is the set $a \le x \le b$ .
The open interval $(a, b)$ is the set $a < x < b$ .
Intervals may also be half-open or half-closed, like $[a, b)$ which is $a \le x < b$ .
The infinite interval $[a, \infty)$ is the set $x \ge a$ .
The infinite interval $(-\infty, a]$ is the set $x \le a$ .

A union of intervals, denoted with $\cup$ , combines two or more sets.

Examples

The inequality $-4 \le x < 1$ is written in interval notation as $[-4, 1)$ .

The set of all numbers greater than 5 is written as the infinite interval $(5, \infty)$ .

Section 3

Absolute Value

Property

The absolute value of $x$ represents the distance from $x$ to the origin on the number line. Because distance is never negative, the absolute value is always non-negative. It is defined piecewise:

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

Examples

To simplify $|3 - 8|$ , first perform the operation inside the bars: $|3 - 8| = |-5|$ . Then take the absolute value, which is $5$ .
In the expression $|3| - |8|$ , we evaluate each absolute value first: $3 - 8$ . The result is $-5$ . This shows that $|a-b|$ is not always the same as $|a|-|b|$ .
To simplify $5 + 2|-4|$ , we first evaluate the absolute value: $|-4| = 4$ . The expression becomes $5 + 2(4) = 5 + 8 = 13$ .

Explanation

Absolute value essentially makes any number positive. It measures a number's distance from zero on a number line, and distance is always a positive concept. Whether a number is positive or negative, its absolute value is its positive counterpart.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Piecewise Function Definition and Cases Notation

Property

A piecewise-defined function is a function that uses different rules (expressions) for different intervals of its domain. The standard notation uses cases:

f(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

Interval Notation

Property

An interval is a set that consists of all the real numbers between two numbers $a$ and $b$ .

The closed interval $[a, b]$ is the set $a \le x \le b$ .
The open interval $(a, b)$ is the set $a < x < b$ .
Intervals may also be half-open or half-closed, like $[a, b)$ which is $a \le x < b$ .
The infinite interval $[a, \infty)$ is the set $x \ge a$ .
The infinite interval $(-\infty, a]$ is the set $x \le a$ .

A union of intervals, denoted with $\cup$ , combines two or more sets.

Examples

The inequality $-4 \le x < 1$ is written in interval notation as $[-4, 1)$ .

The set of all numbers greater than 5 is written as the infinite interval $(5, \infty)$ .

Section 3

Absolute Value

Property

The absolute value of $x$ represents the distance from $x$ to the origin on the number line. Because distance is never negative, the absolute value is always non-negative. It is defined piecewise:

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

Examples

To simplify $|3 - 8|$ , first perform the operation inside the bars: $|3 - 8| = |-5|$ . Then take the absolute value, which is $5$ .
In the expression $|3| - |8|$ , we evaluate each absolute value first: $3 - 8$ . The result is $-5$ . This shows that $|a-b|$ is not always the same as $|a|-|b|$ .
To simplify $5 + 2|-4|$ , we first evaluate the absolute value: $|-4| = 4$ . The expression becomes $5 + 2(4) = 5 + 8 = 13$ .

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Piecewise Function Definition and Cases Notation

Property

A piecewise-defined function is a function that uses different rules (expressions) for different intervals of its domain. The standard notation uses cases:

f(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

Interval Notation

Property

An interval is a set that consists of all the real numbers between two numbers $a$ and $b$ .

The closed interval $[a, b]$ is the set $a \le x \le b$ .
The open interval $(a, b)$ is the set $a < x < b$ .
Intervals may also be half-open or half-closed, like $[a, b)$ which is $a \le x < b$ .
The infinite interval $[a, \infty)$ is the set $x \ge a$ .
The infinite interval $(-\infty, a]$ is the set $x \le a$ .

A union of intervals, denoted with $\cup$ , combines two or more sets.

Examples

The inequality $-4 \le x < 1$ is written in interval notation as $[-4, 1)$ .

The set of all numbers greater than 5 is written as the infinite interval $(5, \infty)$ .

Section 3

Absolute Value

Property

The absolute value of $x$ represents the distance from $x$ to the origin on the number line. Because distance is never negative, the absolute value is always non-negative. It is defined piecewise:

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

Examples

To simplify $|3 - 8|$ , first perform the operation inside the bars: $|3 - 8| = |-5|$ . Then take the absolute value, which is $5$ .
In the expression $|3| - |8|$ , we evaluate each absolute value first: $3 - 8$ . The result is $-5$ . This shows that $|a-b|$ is not always the same as $|a|-|b|$ .
To simplify $5 + 2|-4|$ , we first evaluate the absolute value: $|-4| = 4$ . The expression becomes $5 + 2(4) = 5 + 8 = 13$ .