Grade 8 Workbook — Chapter 7: Relations, Functions, Domain, and Range

SOL 8.PFA.2 · Companion to Textbook Chapter 7

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 104.

PAGE 1 — Chapter opener

Chapter 7 · Relations, Functions, Domain, and Range

Standard 8.PFA.2

In this chapter you will:

Read a relation as a set of ordered pairs, as a table, or as a graph of discrete points
Decide whether a relation is a function
Use the vertical line test on a graph of separate dots
Learn why a repeated output is fine, and why a repeated input is only sometimes a problem
Identify the domain and the range from all three representations
Say what an ordered pair, a domain, and a range mean in a real situation

Words to know: relation · ordered pair · input · output · $x$ -coordinate · $y$ -coordinate · table · discrete points · function · vertical line test · mapping diagram · domain · range

Scope note: every relation in this chapter has at most 10 ordered pairs, and every graph is a set of separate dots. Never connect the dots.

PAGE 2 — Three ways to show one relation

7.1 Relations and Three Ways to Show Them

FIGURE: fig1-three-representations.png (full width)

Fill in the blanks.

A relation is any set of ____________ ____________.

In the ordered pair $(x, y)$ , the first number is the ____________ and the second number is the ____________.

A relation in this chapter has at most ______ ordered pairs.

Discrete means ____________________, so the graph of a relation here is a set of ______________ and never a line.

Write $\{(2,5),(4,7),(6,9)\}$ as a table.

$x$	$y$

A table lists $x$ -values $0, 1, 2, 3$ with $y$ -values $5, 3, 1, -1$ .

As a set of ordered pairs: _______________________________________
In $(7, -2)$ : input ______ output ______

PAGE 3 — Reading a graph

Reading Points Off a Graph

FIGURE: fig3-vertical-line-test-discrete.png (full width)

List the ordered pairs of Graph P as a set.
The largest number of ordered pairs a relation in this chapter may have: ______
Why is the graph of a relation here a set of separate dots rather than a line?
Write each table as a set of ordered pairs.

Table	Set of ordered pairs
a) $x$ : $-1, 0, 2, 5$ and $y$ : $4, 4, -2, 7$
b) $x$ : $3, 3, 6$ and $y$ : $1, 8, 1$

Write $\{(-4,1),(-2,0),(0,-1),(2,-2)\}$ as a table.

$x$	$y$

List the ordered pairs of Graph Q as a set.

PAGE 4 — Building relations

From a Rule to a Relation

Each output is $4$ times its input, and the inputs are $0, 1, 2, 3$ .

As a set: _______________________________________
Write the relation from item 10 as a table.

$x$	$y$

Explain. Which representation makes a repeated input easiest to spot? ____________________

Which makes it easiest to miss? ____________________ Why?
Apply it. A parking garage charges $3 per hour, for $1$ , $2$ , $3$ , and $4$ hours.

As a set: _______________________________________

The input represents ____________________ The output represents ____________________
Find the error. A table lists $x$ : $2, 5$ and $y$ : $7, 9$ . A student writes $\{(7,2),(9,5)\}$ .

What went wrong? _______________________________________________

Correct relation: _______________________

PAGE 5 — Exit ticket 7.1

Exit Ticket · Lesson 7.1

Name: ________________________ Date: ____________

Write $\{(0,3),(1,5),(2,7)\}$ as a table.

$x$	$y$

A table lists $x$ : $-3, -1, 4$ and $y$ : $6, 6, -2$ . As a set: _______________________
In $(-5, 8)$ : input ______ output ______
Why are $(3,8)$ and $(8,3)$ different ordered pairs?

PAGE 6 — The function rule

7.2 Deciding Whether a Relation Is a Function

FIGURE: fig2-mapping-diagrams.png (full width)

Complete the rule.

A function is a relation in which each ____________ is paired with exactly ______ ____________.

A repeated output ____________ breaks the rule. (never / always)

A repeated input breaks the rule only when the two outputs are ____________________.

FIGURE: fig4-repeated-ordered-pair.png (full width)

Fill in the frame for $\{(1,3),(2,5),(1,3),(4,6)\}$ .

The input $1$ appears ______ times. Its output is ______ both times, so the input has ______ output.

Function? ______

Function or not? Give the reason.

$\{(1,3),(2,5),(3,7)\}$ : ______ because _______________________________________
$\{(4,1),(4,6),(5,2)\}$ : ______ because _______________________________________
$\{(2,8),(3,8),(5,8)\}$ : ______ because _______________________________________
$\{(1,2),(2,4),(1,2),(3,6)\}$ : ______ because ___________________________________

PAGE 7 — The vertical line test

The Vertical Line Test on Discrete Points

FIGURE: fig3-vertical-line-test-discrete.png (full width)

Complete the test. If any vertical line passes through ______ or more plotted points, the relation is ______ a function.

Is Graph P a function? ______ Test used: ____________________
Is Graph Q a function? ______ A vertical line that proves it: $x =$ ______
Function or not? Give the reason.

Relation	Function?	Reason
a) $\{(-2,5),(0,5),(3,5),(6,5)\}$
b) $\{(7,1),(8,2),(7,3)\}$
c) $\{(0,0),(1,1),(2,4),(3,9)\}$
d) $\{(-1,2),(-1,2),(4,5)\}$

$x$ : $1, 2, 3, 4, 5$ and $y$ : $10, 8, 6, 8, 10$ . Function? ______ Why?
$x$ : $-2, -1, 0, -1$ and $y$ : $3, 5, 7, 9$ . Function? ______ Why?

PAGE 8 — Sorting four graphs

Four Graphs to Sort

FIGURE: fig6-four-relations-a-to-d.png (full width)

Which of Graphs A, B, C, and D are functions?

Graph	Ordered pairs	Function?	If not, which vertical line?
A
B
C
D

A relation has $10$ ordered pairs and every $x$ -value is different. Must it be a function? ______

Why? _______________________________________________

PAGE 9 — Reasoning and application

Show Why

Explain. Why does a repeated output never break the function rule, but a repeated input sometimes does?

Repeated output example: _______________________ Verdict: ______

Repeated input example: _______________________ Verdict: ______

Explanation: _______________________________________________
Apply it. A dog walker charges $12 per walk: $\{(1,12),(2,24),(3,36),(4,48)\}$ .

Function? ______ Why would a price list that is not a function be a problem for a customer?
Find the error. A student says $\{(2,3),(4,3),(6,3)\}$ is not a function "because $3$ repeats."

What did the student confuse? _______________________________________________

Correct verdict: ______

PAGE 10 — Exit ticket 7.2

Exit Ticket · Lesson 7.2

Name: ________________________ Date: ____________

$\{(5,1),(6,2),(7,3)\}$ : function? ______ Why? _______________________
$\{(0,4),(1,5),(0,6)\}$ : function? ______ Why? _______________________
$\{(3,9),(3,9),(4,16)\}$ : function? ______ Why? _______________________
State the vertical line test for a graph of discrete points, and say what it detects.

PAGE 11 — Domain and range

7.3 Domain and Range

Complete the definitions.

The domain is the set of all ____________ — the ______-values.

The range is the set of all ____________ — the ______-values.

Two rules for writing either set: no ____________, and ____________ order.

FIGURE: fig5-domain-and-range-from-a-graph.png (full width)

Give the domain and range.

$\{(1,4),(2,5),(3,6)\}$ Domain: _______________ Range: _______________
$\{(-3,0),(-1,2),(4,2)\}$ Domain: _______________ Range: _______________
$\{(2,7),(2,7),(5,9)\}$ Domain: _______________ Range: _______________
$x$ : $0, 1, 2, 3$ and $y$ : $-1, -1, 3, 5$ Domain: _______________ Range: _______________
The graph in the figure above Domain: _______________ Range: _______________
Why is a value that appears twice as an output written only once in the range?

PAGE 12 — Domain and range from graphs

Reading Both Sets Off a Graph

FIGURE: fig6-four-relations-a-to-d.png (full width)

Graph A Domain: _______________ Range: _______________
Graph C Domain: _______________ Range: _______________
Graph B (not a function, but the two sets can still be listed)

Domain: _______________ Range: _______________
Graph D Domain: _______________ Range: _______________

PAGE 13 — Practice · domain and range

Practice · Domain and Range

Give the domain and range.

Relation	Domain	Range
a) $\{(-5,2),(-4,4),(0,6),(3,8)\}$
b) $\{(1,-3),(2,-3),(3,-3)\}$
c) $\{(-2,9),(0,7),(-2,9),(5,1)\}$
d) $\{(6,0),(4,1),(2,2),(0,3)\}$

$x$ : $5, 10, 15, 20$ and $y$ : $2.5, 5, 7.5, 10$ Domain: _______________ Range: _______________
A function has domain $\{1,2,3,4\}$ and every output is twice its input.

Ordered pairs: _______________________________ Range: _______________
A relation has domain $\{-1,0,1\}$ and range $\{5\}$ .

Ordered pairs: _______________________ Function? ______

PAGE 14 — Reasoning and application

How Big Can Each Set Be?

Explain. Can the range of a function contain fewer values than its domain? ______

Example or reason: _______________________________________________
Explain. Can the range of a function contain more values than its domain? ______

Example or reason: _______________________________________________
Apply it. $\{(0,50),(1,45),(2,40),(3,35)\}$ pairs minutes read with pages left.

Domain: _______________ Range: _______________

The domain means _______________________ The range means _______________________
Find the error. A student gives the range of $\{(1,3),(2,3),(4,5)\}$ as $\{3,3,5\}$ .

What went wrong? _______________________________________________

Correct range: _______________

PAGE 15 — Exit ticket 7.3

Exit Ticket · Lesson 7.3

Name: ________________________ Date: ____________

$\{(-4,1),(0,3),(2,3),(6,7)\}$ Domain: _______________ Range: _______________
$x$ : $8, 6, 4, 2$ and $y$ : $0, -2, -4, -6$ Domain: _______________ Range: _______________
$\{(3,3),(3,3),(9,1)\}$ Domain: _______________ Range: _______________
Explain the two conventions for writing a domain or a range.

PAGE 16 — The three-step routine

7.4 Putting It All Together

The routine.

Step 1: What are the ____________ ____________?

Step 2: Is it a ____________? Look for one ____________ with two different ____________.

Step 3: What are the ____________ and the ____________?

FIGURE: fig7-bracelet-cost-discrete-points.png (full width)

Ordered pairs shown: _______________________________________
Function? ______ Why? _______________________________________
Domain: _______________ Range: _______________
What does $(3,12)$ mean here? _______________________________________
$x$ : $1, 2, 2, 3$ and $y$ : $6, 7, 9, 10$ Function? ______ Why? _______________________
$\{(-1,-1),(0,0),(1,1),(2,2)\}$ Function? ______ Domain: _____________ Range: _____________

PAGE 17 — Practice · all three questions

Practice · Function, Domain, Range

Complete the table.

Relation	Function?	Domain	Range
a) $\{(2,1),(4,2),(6,3),(8,4)\}$
b) $\{(-3,5),(-3,-5),(0,0)\}$
c) $\{(1,7),(1,7),(2,9),(3,9)\}$

Hourly temperatures — $x$ : $1, 2, 3, 4, 5$ and $y$ : $62, 65, 68, 65, 62$

Function? ______ Domain: _______________ Range: _______________
Graph D is not a function. Vertical line that proves it: $x =$ ______

The two points it passes through: _______________ and _______________
Apply it. A taxi charges $3 plus $2 per mile: $\{(1,5),(2,7),(3,9),(4,11)\}$ .

Function? ______ Domain: _______________ Range: _______________

$(4,11)$ means _______________________________________

PAGE 18 — Reasoning and error hunt

Think It Through

A machine multiplies each input by itself. Inputs: $-2, -1, 0, 1, 2$ .

Ordered pairs: _______________________________________

Function? ______ Range: _______________
Write a relation with four ordered pairs that is not a function.

Now change exactly one number to make it a function: _______________________________________

Why does your change work? _______________________________________________
Explain. Why is a vertical line meeting two plotted points the same thing as one input having two outputs?
Find the error. A student says $\{(1,2),(1,2)\}$ is not a function because the input $1$ appears twice.

What went wrong? _______________________________________________ Correct verdict: ______
Find the error. A student gives the domain of $\{(4,1),(2,3)\}$ as $\{1,3\}$ .

What went wrong? _______________________________________________

Correct domain: _______________ Correct range: _______________
Apply it. $\{(1,0.5),(2,1.5),(3,1.5),(4,0)\}$ pairs the day with inches of rain.

Function? ______ Domain: _______________ Range: _______________

PAGE 19 — Exit ticket 7.4

Exit Ticket · Lesson 7.4

Name: ________________________ Date: ____________

$\{(0,2),(3,4),(6,2),(9,8)\}$ Function? ______ Domain: _____________ Range: _____________
$x$ : $-2, -2, 0, 2$ and $y$ : $1, 5, 5, 7$ Function? ______ Domain: _____________ Range: _____________
What does $(5,20)$ mean in the bracelet situation? _______________________________________
Describe the three-step routine of this chapter.

PAGE 20 — Chapter 7 review, part 1

Chapter 7 Review

Part A · Is it a function?

$\{(1,1),(2,2),(3,3)\}$ : ______ Why? _______________________
$\{(-5,2),(-5,3)\}$ : ______ Why? _______________________
$\{(0,6),(1,6),(2,6),(3,6)\}$ : ______ Why? _______________________
$\{(8,2),(8,2),(9,4)\}$ : ______ Why? _______________________
$x$ : $1, 3, 5, 7$ and $y$ : $2, 4, 2, 4$ : ______ Why? _______________________
$x$ : $2, 4, 4, 6$ and $y$ : $1, 3, 5, 7$ : ______ Why? _______________________

PAGE 21 — Chapter 7 review, part 2

Chapter 7 Review (continued)

FIGURE: fig6-four-relations-a-to-d.png (full width)

Is Graph A a function? ______ Why? _______________________________________
Is Graph B a function? ______ Vertical line that proves it: $x =$ ______
State the function rule in your own words, and explain why a repeated ordered pair does not break it.
Explain. A relation is written with $10$ ordered pairs but only $6$ different $x$ -values. Can it still be a function? ______

What would have to be true? _______________________________________________

PAGE 22 — Chapter 7 review, part 3

Chapter 7 Review (continued)

Part B · Domain and range

$\{(-2,4),(1,5),(3,6)\}$ Domain: _______________ Range: _______________
$\{(0,-2),(2,-2),(4,-2)\}$ Domain: _______________ Range: _______________
$\{(5,1),(5,1),(7,3)\}$ Domain: _______________ Range: _______________
$x$ : $10, 20, 30$ and $y$ : $0.5, 1, 1.5$ Domain: _______________ Range: _______________

FIGURE: fig5-domain-and-range-from-a-graph.png (half width)

The graph above Domain: _______________ Range: _______________
Graph C Domain: _______________ Range: _______________
Graph D Domain: _______________ Range: _______________
A function has domain $\{-3,0,3\}$ and every output is one less than its input.

Ordered pairs: _______________________ Range: _______________
Write a function with domain $\{1,2,3,4\}$ whose range has exactly two values.
Explain. Why can the range of a function never contain more values than its domain?

PAGE 23 — Chapter 7 review, part 4

Chapter 7 Review (continued)

Part C · Application and reasoning

Apply it. $\{(1,3),(2,6),(3,9),(4,12),(5,15)\}$ pairs cups of flour with cookies.

Function? ______ Domain: _______________ Range: _______________

$(4,12)$ means _______________________________________
The bracelet relation $\{(1,4),(2,8),(3,12),(4,16),(5,20)\}$ .

Function? ______ Domain: _______________ Range: _______________

Could $(3,15)$ be added and leave it a function? ______ Why?
Find the error. A graph shows dots at $(2,1)$ , $(2,5)$ , and $(4,3)$ . A student says it is a function "because the two points are different points."

What went wrong? _______________________________________________ Correct verdict: ______
Explain. Why does the vertical line test work?
Write a table for a function with $5$ ordered pairs, domain $\{1,2,3,4,5\}$ , and a range with exactly $3$ values.

$x$	$1$	$2$	$3$	$4$	$5$
$y$

 Range: _______________

Apply it. A shop charges a $3 delivery fee plus $9 per pizza, for $1$ to $4$ pizzas.

As a set: _______________________________________

Function? ______ Domain: _______________ Range: _______________

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Item order: pages 12 and 13 split the domain-and-range set by representation rather than by number, so items 47 and 48 (graphs) sit beside items 43 and 44 while items 45, 46, 49, and 50 (sets and tables) share the next page. The numbers still match the textbook exactly.
Figure widths: fig1, fig2, fig3, fig4, fig6, and fig7 are full width. fig6-four-relations-a-to-d.png is square and tall — give it its own block with no text beside it, and repeat it on pages 8, 12, and 21 rather than asking students to flip back.
Blank coordinate grids: where a student is asked to graph, supply a printed grid from $-5$ to $5$ on both axes with labeled unit ticks. Students plot dots only — add the reminder "plot dots; do not connect them" beside every grid.
Set braces: set every answer blank for a domain, a range, or a set of ordered pairs wide enough for braces and commas on one line, so the Canva text box does not reflow mid-set.
Negative signs: use the typographic minus (−), matching the figures, not a hyphen.