Algebra 1 Workbook — Chapter 4: Functions, Domain, and Range

SOL A.F.2 (a) · Companion to Textbook Chapter 4

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 116.

PAGE 1 — Chapter opener

Chapter 4 · Functions, Domain, and Range

Standard A.F.2 (a)

In this chapter you will:

• Read a relation as a set of ordered pairs, a table, a mapping, or a graph
• Decide whether a relation is a function, in all four representations
• Use the vertical line test on a graph of dots and on an unbroken curve
• Learn the two traps: a repeated output is always fine, a repeated input only sometimes
• State the domain and the range — as a listed set, or in words for a curve
• Read $f(x)$ as the name of a rule and the name of its output

Words to know: relation · ordered pair · input · output · table · mapping · graph · discrete · continuous · function · vertical line test · $f(x)$ · domain · range

Convention: a domain or a range for a relation with finitely many pairs is written in braces, each value once, in increasing order. For an unbroken curve, describe both sets in words.

PAGE 2 — Four ways to show one relation

4.1 Relations and Their Four Representations

FIGURE: fig1-four-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 is the ____________.

The four representations this standard names are:

____________________ · ____________________ · ____________________ · ____________________

1. Write $\{(-3,4),(0,1),(2,-5)\}$ as a table.

2. A table lists $x$-values $-2, 1, 4, 7$ with $y$-values $0, 3, 6, 9$.

   As a set of ordered pairs: _______________________________________

3. In $(-6, 11)$: input ______ output ______

PAGE 3 — Discrete and continuous

Two Kinds of Graph

FIGURE: fig5-discrete-and-continuous-graphs.png (full width)

Fill in the blanks.

A discrete graph is a set of ____________________ points.

A continuous graph is an ____________________ curve or line.

6. In one sentence each, say what a discrete graph is and what a continuous graph is.

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7. List, as a set of ordered pairs, the four points on the discrete graph above.

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8. Name the four representations this standard requires.

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PAGE 4 — Mappings and rules

From a Rule to a Relation

FIGURE: fig9-three-mappings-to-sort.png (full width)

4. List the ordered pairs of Mapping M as a set.

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5. Describe the mapping of $\{(2,5),(3,5),(6,1)\}$.

   Left box: _______________ Right box: _______________ Number of arrows: ______

6. Write each table as a set of ordered pairs.

8. Write $\{(-4,-4),(-2,0),(1,3),(5,-1)\}$ as a table.

11. Each output is three less than twice its input; the inputs are $-1$, $0$, $2$, $5$.

    As a set: _______________________________________

12. Write the relation from item 11 as a table.

PAGE 5 — Reasoning and application

Show Why

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

    Which makes it easiest to miss? ____________________ Why?

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14. Apply it. A craft booth rents for $6 per hour, for $1$, $2$, $3$, and $4$ hours.

    As a set: _______________________________________

    The input represents ____________________ The output represents ____________________

15. Find the error. A table lists $x$: $3, 8$ and $y$: $-1, 5$. A student writes $\{(-1,3),(5,8)\}$.

    What went wrong? _______________________________________________

    Correct relation: _______________________

16. Explain. $\{(1,2),(2,1)\}$ and $\{(2,1),(1,2)\}$ are the same relation, but $(1,2)$ and $(2,1)$ are different pairs. Why are both statements true?

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PAGE 6 — Exit ticket 4.1

Exit Ticket · Lesson 4.1

Name: ________________________ Date: ____________

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

18. A table lists $x$: $1, 2, 3$ and $y$: $-4, -8, -12$. As a set: _______________________

19. In $(9, -2)$: input ______ output ______

20. Name the four representations, and how you read a relation out of each.

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PAGE 7 — The function rule

4.2 Deciding Whether a Relation Is a Function

FIGURE: fig2-mapping-function-and-not.png (full width)

Complete the definition.

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

In a mapping, count the arrows that ____________ each input, never the arrows that ____________ at an output.

FIGURE: fig3-mapping-many-to-one.png (half width)

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

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

Fill in the frame for $\{(-1,3),(6,0),(-1,3),(2,8)\}$.

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

Function? ______

Function or not? Give the reason.

21. $\{(-3,2),(0,5),(4,9)\}$: ______ because _______________________________________

22. $\{(1,6),(1,9),(4,2)\}$: ______ because _______________________________________

23. $\{(2,7),(5,7),(8,7)\}$: ______ because _______________________________________

24. $\{(-1,3),(6,0),(-1,3),(2,8)\}$: ______ because ___________________________________

PAGE 8 — Function notation

Naming a Function: $f(x)$

Complete the frame.

In $f(x)$, the letter $f$ names the ____________. The whole symbol $f(x)$ names the ____________ at the input $x$.

Read aloud: "____________________." It is ______ multiplication. (is / is not)

The statement $f(4) = 2$ is the same fact as the ordered pair ____________.

25. In the two-mapping figure, which one is a function? ______ What did you count?

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26. In the many-to-one figure, is the relation a function? ______

    Explain using the words leave and arrive: _______________________________________

27. Explain. The function $\{(0,0),(1,1),(4,2),(9,3)\}$ is named $f$. What does $f(4) = 2$ say?

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    Why could $\{(1,6),(1,9)\}$ not be named this way? _______________________________________

PAGE 9 — Practice · sets, tables, mappings

Practice · Is It a Function?

27. Decide and give the reason.

28. $x$: $2, 4, 6, 8, 10$ and $y$: $15, 12, 9, 12, 15$. Function? ______ Why?

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29. $x$: $-4, -2, -4, 1$ and $y$: $6, 6, 9, 0$. Function? ______ Why?

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30. $x$: $5, 5, 5$ and $y$: $2, 2, 2$. Function? ______ Domain: __________ Range: __________

31. FIGURE: fig9-three-mappings-to-sort.png (full width)

PAGE 10 — Reasoning and application

Think It Through

32. A relation is written with $6$ ordered pairs, but only $4$ different inputs appear. Can it be a function? ______

    What would have to be true? _______________________________________________

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

    Repeated output example: _______________________ Verdict: ______

    Repeated input example: _______________________ Verdict: ______

    Explanation: _______________________________________________

34. Apply it. Lockers: $\{(214,5),(215,9),(216,5)\}$ pairs a student ID with a locker number.

    Function? ______ $(216,5)$ says _______________________________________

    Why would containing both $(214,5)$ and $(214,9)$ be a problem?

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35. Find the error. A student says $\{(3,8),(5,8),(9,8)\}$ is not a function "because $8$ repeats."

    What did the student confuse? _______________________________________________

    Correct verdict: ______

PAGE 11 — Exit ticket 4.2

Exit Ticket · Lesson 4.2

Name: ________________________ Date: ____________

37. $\{(7,1),(8,2),(9,3)\}$: function? ______ Why? _______________________

38. $\{(0,5),(2,6),(0,9)\}$: function? ______ Why? _______________________

39. $\{(-4,2),(-4,2),(6,2)\}$: function? ______ Why? _______________________

40. What do the two parts of $f(x)$ name, and why can only a function be written this way?

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PAGE 12 — The vertical line test

4.3 Graphs and the Vertical Line Test

FIGURE: fig4-vertical-line-test.png (full width)

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

Two special lines. A horizontal line such as $y = 4$ ______ a function. A vertical line such as $x = -3$ ______ a function. (is / is not)

41. Is the graph on the left a function? ______ Test used: ____________________

42. Is the circle a function? ______ A vertical line that proves it: $x =$ ______

    The two points it passes through: _______________ and _______________

PAGE 13 — Sorting four graphs

Four Graphs to Sort

FIGURE: fig8-four-graphs-a-to-d.png (full width)

43. Graph A — function? ______ Ordered pairs: _______________________________

44. Graph B — function? ______ Vertical line that proves it: $x =$ ______

45. Graph C — function? ______ Why? _______________________________________

46. Graph D — function? ______ Vertical line that proves it: $x =$ ______

PAGE 14 — Practice · the test

Practice · Passing and Failing

47. Points $(-5,4)$, $(-2,1)$, $(0,1)$, $(3,-6)$. Function? ______ Why? ______________________

48. Points $(1,2)$, $(1,-2)$, $(4,0)$. Function? ______ Vertical line: $x =$ ______

49. The vertical line $x = -3$. Function? ______ Why? _______________________________

50. The horizontal line $y = 4$. Function? ______ Why? _______________________________

51. Which fail the test?

PAGE 15 — Reasoning and application

Why the Test Works

51. Explain. Why does the vertical line test work? What do two points on one vertical line have in common?

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52. Apply it. The ticket-cost graph (page 20). Function? ______

    What would a "no" answer have meant for a customer? _______________________________________

53. Find the error. A student says $y = 4$ is not a function "because every input gives the same output, so the outputs repeat."

    What went wrong? _______________________________________________ Correct verdict: ______

54. Explain. A graph passes the test. Can two of its points share the same $y$-coordinate? ______

    Example: _______________________________________

55. Apply it. A weather station plots the temperature once a minute against the time. Why must that graph pass the test?

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PAGE 16 — Exit ticket 4.3

Exit Ticket · Lesson 4.3

Name: ________________________ Date: ____________

57. Points $(0,3)$, $(2,5)$, $(0,-1)$. Function? ______ Vertical line: $x =$ ______

58. State the vertical line test and say what it detects.

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59. Is the unbroken line $y = x - 1$ a function? ______ Why? _______________________

60. Why can a circle never be the graph of a function?

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PAGE 17 — Domain and range

4.4 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.

For an unbroken curve, describe both sets in ____________.

FIGURE: fig6-domain-and-range-discrete.png (full width)

Give the domain and range.

61. $\{(-3,4),(1,0),(6,4)\}$ Domain: _______________ Range: _______________

62. $\{(2,9),(2,9),(5,-1)\}$ Domain: _______________ Range: _______________

63. $x$: $-6, -2, 0, 3$ and $y$: $5, 5, -4, 7$ Domain: _______________ Range: _______________

64. Mapping M Domain: _______________ Range: _______________

65. The graph above Domain: _______________ Range: _______________

PAGE 18 — Domain and range from a curve

Reading Both Sets Off a Curve

FIGURE: fig7-domain-and-range-continuous.png (full width)

Endpoint frame. A filled-in point means the endpoint ______ included. An open point means the endpoint ______ included. (is / is not)

66. The line $y = 2x - 1$ Domain: _______________________ Range: _______________________

67. The segment with a filled-in point at $(-2,4)$ and an open point at $(3,-1)$

    Domain: _______________________________________

    Range: _______________________________________

68. Graph C, the line $y = -x + 2$

    Domain: _______________________ Range: _______________________

69. Explain. How does reading the domain off a discrete graph differ from reading it off an unbroken line?

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PAGE 19 — Practice · domain and range

Practice · Domain and Range

67. Give both sets.

68. $x$: $12, 9, 6, 3$ and $y$: $1.5, 1, 0.5, 0$ Domain: _______________ Range: _______________

69. Mapping N (not a function; list both sets anyway) Domain: __________ Range: __________

70. Mapping P Domain: _______________ Range: _______________

71. Graph A Domain: _______________ Range: _______________

72. Graph B (not a function; list both sets anyway) Domain: __________ Range: __________

PAGE 20 — Application

Domain in a Real Situation

FIGURE: fig10-ticket-cost-in-context.png (full width)

75. Apply it. Concert tickets at $8 each, up to six.

    Ordered pairs: _______________________________________

    Function? ______ Domain: _______________ Range: _______________

    $(4,32)$ means _______________________________________

76. Apply it. A café sells muffins for $2.25 each, at most four.

    As a set: _______________________________________

    Function? ______ Domain: _______________ Range: _______________

    Why is the domain not "all real numbers from $1$ to $4$"?

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77. Explain. Can the range of a function with finitely many pairs hold more values than its domain? ______

    Reason: _______________________________________________

78. Find the error. A student gives the domain of $\{(5,2),(1,8)\}$ as $\{2,8\}$.

    What went wrong? _______________________________________________

    Correct domain: _______________ Correct range: _______________

PAGE 21 — Exit ticket 4.4

Exit Ticket · Lesson 4.4

Name: ________________________ Date: ____________

79. $\{(-8,3),(0,3),(4,-2),(4,-2)\}$ Function? ______ Domain: __________ Range: __________

80. $x$: $1, 3, 5$ and $y$: $-2, -2, 6$ Domain: _______________ Range: _______________

81. The discrete graph of Lesson 4.4 Domain: _______________ Range: _______________

PAGE 22 — Chapter 4 review · sets of ordered pairs

Chapter 4 Review

Part A · Relations given as a set of ordered pairs

83. $\{(1,4),(2,8),(3,12)\}$ Function? ______ Domain: __________ Range: __________

84. $\{(-2,6),(-2,-6),(5,0)\}$ Function? ______ Domain: __________ Range: __________

85. $\{(0,7),(4,7),(9,7)\}$ Function? ______ Domain: __________ Range: __________

86. $\{(6,1),(6,1),(8,3)\}$ Function? ______ Domain: __________ Range: __________

87. $\{(-1,-1),(0,0),(1,1),(2,2)\}$ Function? ______ Domain: __________ Range: __________

88. $\{(3,5),(4,5),(3,9)\}$ Function? ______ Domain: __________ Range: __________

89. Write a relation with four ordered pairs that is not a function.

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    Change exactly one number to make it a function: _______________________________________

    Why does your change work? _______________________________________________

90. Explain. A relation has $8$ pairs but only $5$ different inputs. Can it be a function? ______

    What must be true of the three extra listings? _______________________________________

PAGE 23 — Chapter 4 review · tables

Chapter 4 Review (continued)

Part B · Relations given as a table

91. $x$: $2, 4, 6, 8$ and $y$: $5, 5, 5, 5$ Function? ______ Domain: __________ Range: __________

92. $x$: $-3, -1, -3, 2$ and $y$: $4, 8, 4, 0$ Function? ______ Domain: __________ Range: __________

93. $x$: $0, 1, 2, 1$ and $y$: $9, 7, 5, 3$ Function? ______ Domain: __________ Range: __________

94. $x$: $10, 20, 30, 40$ and $y$: $2.5, 5, 7.5, 10$ Function? ______ Domain: __________ Range: __________

95. Apply it. Hours worked and pay — $x$: $1, 2, 3, 4$ and $y$: $15, 30, 45, 60$

    Function? ______ Domain: _______________ Range: _______________

    $(3,45)$ means _______________________________________

96. Explain. How do you check a table for the function property in one pass down the input column? What do you do when you meet a repeat?

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PAGE 24 — Chapter 4 review · mappings

Chapter 4 Review (continued)

Part C · Relations given as a mapping

FIGURE: fig9-three-mappings-to-sort.png (full width)

97. Mapping M Function? ______ Domain: _______________ Range: _______________

98. Mapping N Function? ______ Input that breaks it: ______ Its two outputs: ______ and ______

    Domain: _______________ Range: _______________

99. Mapping P Function? ______ Domain: _______________ Range: _______________

FIGURE: fig3-mapping-many-to-one.png (half width)

100. Is this mapping a function? ______

     Difference between an arrow that leaves and one that arrives:

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101. Describe a mapping with domain $\{-2,0,3\}$ and range $\{7\}$.

     Left box: _______________ Right box: _______________ Arrows: ______ Function? ______

102. Explain. How do you decide the function question from a mapping at a glance, and why do arriving arrows tell you nothing?

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PAGE 25 — Chapter 4 review · graphs

Chapter 4 Review (continued)

Part D · Relations given as a graph

FIGURE: fig8-four-graphs-a-to-d.png (full width)

103. Graph A Function? ______ Domain: _______________ Range: _______________

104. Graph B Function? ______ Vertical line: $x =$ ______

     Domain: _______________ Range: _______________

105. Graph C Function? ______ Domain: _______________________ Range: _______________________

106. Graph D Function? ______ Vertical line: $x =$ ______

107. Explain. Why is the line in the vertical-line-test figure a function? Refer to what every vertical line does.

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108. The segment with a filled-in point at $(-2,4)$ and an open point at $(3,-1)$.

     Domain: _______________________________________

     Range: _______________________________________

     What does the open point change? _______________________________________

109. Apply it. The ticket-cost graph. Function? ______ Domain: __________ Range: __________

     $(6,48)$ means _______________________________________

110. Explain. Why is the vertical line test not an extra rule but the definition in a picture?

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PAGE 26 — Chapter 4 review · mixed

Chapter 4 Review (continued)

Part E · Application and reasoning

111. Apply it. A bike shop charges a $5 deposit plus $4 per hour, for $1$ to $4$ hours.

     As a set: _______________________________________

     Function? ______ Domain: _______________ Range: _______________

     $(3,17)$ means _______________________________________

112. Explain. The two traps, in your own words.

     A repeated output never breaks the rule because _______________________________________

     A repeated input breaks it only when _______________________________________

113. Find the error. A student says $\{(2,3),(2,3)\}$ is not a function because the input $2$ appears twice.

     What went wrong? _______________________________________________

     Correct verdict: ______ Domain: __________ Range: __________

114. Explain. The same relation is given once as a table and once as a graph. Why must the input-column check and the vertical line test always agree?

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115. Write a function with domain $\{1,2,3,4\}$ whose range has exactly two values.

 Range: _______________  Why is it a function? _______________________________________

116. Apply it. The ticket-cost function is named $c$. What does $c(5) = 40$ say?

     The $5$ represents _______________________ The $40$ represents _______________________

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: a few items are placed by representation rather than by number, so that every item on a page can be answered from the figure on that page. Page 3 runs 6, 9, 5; page 4 runs 4, 10, 7, 8, 11, 12; page 14 places 52 after 50; page 18 groups 66, 74, 73, and 82 with the continuous-graph figure. The numbers still match the textbook exactly, and the answer key is in numerical order.
• Figure widths: fig1, fig2, fig4, fig5, fig6, fig7, fig8, fig9, and fig10 are full width. fig3-mapping-many-to-one.png is small and square — set it half width with text beside it. fig8-four-graphs-a-to-d.png is square and tall; give it its own block with no text beside it, and repeat it on pages 13, 19, and 25 rather than asking students to flip back. fig9-three-mappings-to-sort.png repeats on pages 4, 9, and 24.
• Blank work space: items 13, 16, 33, 51, 96, 102, 110, 112, and 114 are written explanations — give each at least four ruled lines, not a single blank. Items 34, 53, 56, 76, and 111 are applications with two or three parts; leave two lines per part.
• Blank coordinate grids: where a student is asked to plot, supply a printed grid from $-6$ to $6$ on both axes with labeled unit ticks. Print two reminders beside every grid: "for a discrete relation, plot dots only — do not connect them," and "for a continuous relation, draw the line all the way across, with arrowheads if it has no endpoints."
• Open and filled points: the endpoint frame on page 18 and item 108 both depend on the reader distinguishing a hollow circle from a solid one. Do not shrink fig7 below half a page, and do not recolor the hollow point's interior.
• Set braces: make every 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. Items 66, 73, 74, 105, and 108 are answered in words, not braces — give those blanks a full line each.
• Negative signs: use the typographic minus (−), matching the figures, not a hyphen.
• Currency: dollar amounts appear on pages 5, 20, and 26. Set them as plain text ($6, $8, $2.25, $5, $4), not inside a math box.