Integrated Math 2–3 Honors · Lesson Review · August 18, 2026

Functions, Domain and Range, and Absolute-Value Equations

Recognize functions, read domain and range, interpret absolute value, and solve equations by splitting into two cases.

25-minute edited lesson10 chaptersCaptions + transcript

01 · Watch

Lesson video

0:0001What Is a Function?1:1302Functions from Ordered Pairs 2:2403Graphs and the Vertical-Line Test 5:0804Domain and Range 8:3105Domain and Range of Common Graphs 10:3106Absolute Value and Its Piecewise Definition 15:5007Solve a Basic Absolute-Value Equation 18:4908Isolate the Absolute Value First 21:1809Expressions Inside Absolute Value 22:3710A Multi-Step Absolute-Value Equation

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02 · Understand

Learning targets

Decide whether a relation is a function from ordered pairs or a graph.
Find the domain and range of a relation.
Interpret absolute value as distance from zero.
Use the piecewise definition of absolute value.
Solve absolute-value equations by isolating the absolute value and writing two equations.

03 · Vocabulary

Key definitions

Use these terms to explain why a relation is or is not a function.

01Relation: A set of ordered pairs.
02Function: A relation in which each input (x-coordinate) corresponds to exactly one output (y-coordinate).
03Vertical-line test: A graph represents a function exactly when every vertical line touches it at most once.
04Domain: The set of all inputs or x-coordinates in a relation.
05Range: The set of all outputs or y-coordinates in a relation.
06Absolute value: A number’s distance from zero; the result is always nonnegative.
07Piecewise-defined function: A function described by different formulas on different parts of its domain.
08Solution set: The set containing every value that makes an open equation true.

04 · Apply

Worked examples

Follow the input-output rule first, then the algebraic steps.

Ordered pairs

Which relation is a function?

\begin{aligned}A&=\{(1,3),(4,6),(5,3)\}\\C&=\{(2,5),(2,8),(6,3)\}\end{aligned}

A is a function: no input repeats with a different output. C is not: the input 2 corresponds to both 5 and 8.

Vertical lines

Test a graph

A nonvertical line passes the vertical-line test. A circle fails because some vertical lines touch it twice.

Domain and range

Read inputs and outputs

Relation	Domain	Range
$A$	$\{1,4,5\}$	$\{3,6\}$
$y=x+5$	$(-\infty,\infty)$	$(-\infty,\infty)$
$y=4$	$(-\infty,\infty)$	$\{4\}$
$y=x^2$	$(-\infty,\infty)$	$[0,\infty)$

Piecewise definition

Define absolute value

|x|=\begin{cases}x,&x\ge0,\\-x,&x<0.\end{cases}

If x is already nonnegative, keep it. If x is negative, its opposite is positive.

Basic equation

Solve $|x|=7$

$x=7$
$x=-7$
Solution set: $\{-7,7\}$

Isolate first

Solve $2|x|-4=6$

$2|x|=10$ add 4
$|x|=5$ divide by 2
$x=\pm5$ write the two cases

Expression inside

Solve $|x-5|=7$

$x-5=7$ or $x-5=-7$
$x=12$ or $x=-2$
Solution set: $\{-2,12\}$

Multi-step

Solve $2|5x-10|=2$

$|5x-10|=1$
$5x-10=1$ or $5x-10=-1$
$x=\frac{11}{5}$ or $x=\frac{9}{5}$

05 · Avoid

Common mistakes

Repeated outputs are allowed.

A function may send different inputs to the same output.

Repeated inputs need care.

The same input cannot correspond to two different outputs.

Domain is not always “all real numbers.”

Look left-to-right or list the x-coordinates actually used.

Absolute value is not always $x$ .

For negative $x$ , $|x|=-x$ .

Isolate before splitting.

Get the absolute-value expression alone before writing the positive and negative cases.

Keep both solutions.

Unless one fails when checked, include both values in the solution set.

06 · Check yourself

Quick check

Decide first, then open the answer.

1Is

\{(1,4),(2,4),(3,7)\}

a function?

Yes. Each input appears only once; repeated outputs are allowed.

2What is the domain of

\{(-2,5),(0,8),(4,5)\}

$\{-2,0,4\}$ . The domain is the set of x-coordinates.

3What is the range of

y=4

$\{4\}$ . Every point has the same output, 4.

4Solve

|x|=9

$x=\pm9$ . The solution set is $\{-9,9\}$ .

5Solve

|x+2|=5

$x=3$ or $x=-7$ . Solve $x+2=5$ and $x+2=-5$ .

07 · Revisit

Transcript and assigned practice

Search the corrected transcript for a definition or example. Then complete ID 1 on the eKadence worksheet Solving Absolute Value Equations in 1 Variable and the assigned DeltaMath practice.

Download corrected transcript