Appendix A — Answer Key, Chapter 7: Relations, Functions, Domain, and Range

SOL 8.PFA.2 · Covers textbook Chapter 7 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 104 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Conventions used in every answer below: a domain or a range is written inside braces, with each value listed once and in increasing order. A relation is a function exactly when no input is paired with two different outputs — a repeated output never breaks the rule, and an input listed twice with the same output does not break it either.

The four relations used repeatedly in the chapter, for reference:

• Graph P $= \{(-3,2),(-1,4),(1,1),(3,3)\}$ and Graph Q $= \{(-2,1),(-2,4),(0,2),(2,5)\}$
• Graph A $= \{(-3,1),(-1,3),(2,3),(4,0)\}$, Graph B $= \{(-2,-1),(1,2),(1,-3),(3,4)\}$, Graph C $= \{(-4,2),(-2,2),(0,2),(2,2)\}$, Graph D $= \{(0,0),(0,3),(2,1),(4,-2)\}$
• Bracelets $= \{(1,4),(2,8),(3,12),(4,16),(5,20)\}$

Lesson 7.1 — Relations and Three Ways to Show Them

Guided practice

1. $x$	$y$
   $2$	$5$
   $4$	$7$
   $6$	$9$

2. $\{(0,5),(1,3),(2,1),(3,-1)\}$
3. Input $7$, output $-2$.
4. $\{(-3,2),(-1,4),(1,1),(3,3)\}$
5. Ten.
6. The relation lists only finitely many ordered pairs, so it states a fact at each listed input and says nothing in between. Connecting the dots would claim outputs at inputs the relation never mentioned — for example an output at $x = 2.5$ when the relation only listed $x = 1, 2, 3, 4$.

Independent practice

7. a) $\{(-1,4),(0,4),(2,-2),(5,7)\}$ b) $\{(3,1),(3,8),(6,1)\}$

8. $x$	$y$
   $-4$	$1$
   $-2$	$0$
   $0$	$-1$
   $2$	$-2$

9. $\{(-2,1),(-2,4),(0,2),(2,5)\}$
10. $\{(0,0),(1,4),(2,8),(3,12)\}$

11. $x$	$y$
    $0$	$0$
    $1$	$4$
    $2$	$8$
    $3$	$12$

12. The table makes a repeated input easiest to spot, because the inputs sit in one column and a value appearing twice is directly visible. A graph can hide it in one specific way — if the same ordered pair is listed twice, the two dots land on top of each other and you see only one — although a repeated input with two different outputs shows up clearly on a graph as two dots on the same vertical line. A set of ordered pairs is the easiest to scan carelessly, since the inputs are scattered through the list.
13. $\{(1,3),(2,6),(3,9),(4,12)\}$. The input is the number of hours parked; the output is the cost in dollars.
14. The student reversed the coordinates, writing each output first. In an ordered pair the input is always written first, and a table pairs each $x$ with the $y$ beside it. Correct relation: $\{(2,7),(5,9)\}$.

Exit ticket 7.1

15. $x$	$y$
    $0$	$3$
    $1$	$5$
    $2$	$7$

16. $\{(-3,6),(-1,6),(4,-2)\}$
17. Input $-5$, output $8$.
18. The pair is ordered, so position carries meaning: the first number is the input and the second is the output. In $(3,8)$ the input is $3$ and the output is $8$; in $(8,3)$ the input is $8$ and the output is $3$. Those are two different statements, and they plot as two different points.

Lesson 7.2 — Deciding Whether a Relation Is a Function

Guided practice

19. Yes. The inputs $1$, $2$, and $3$ are all different, so no input can have two outputs.
20. No. The input $4$ is paired with $1$ and also with $6$, which is two different outputs for one input.
21. Yes. Each of the inputs $2$, $3$, and $5$ appears once, so each has exactly one output. A repeated output is allowed.
22. Yes. The input $1$ appears twice, but both times its output is $3$, so the input $1$ still has exactly one output.
23. Yes, Graph P is a function, by the vertical line test: every vertical line passes through at most one of the four dots.
24. No, Graph Q is not a function. The vertical line $x = -2$ passes through both $(-2,1)$ and $(-2,4)$.

Independent practice

25. a) Function. The inputs $-2$, $0$, $3$, $6$ are all different; the repeated output $5$ does not matter. b) Not a function. The input $7$ has outputs $1$ and $3$. c) Function. The inputs $0$, $1$, $2$, $3$ each appear once. d) Function. The pair $(-1,2)$ is listed twice, so the input $-1$ has the single output $2$.
26. Function. The inputs $1, 2, 3, 4, 5$ each appear once. The outputs $8$ and $10$ each repeat, which the rule allows.
27. Not a function. The input $-1$ appears in two rows, once with the output $5$ and once with the output $9$.
28. Graphs A and C are functions; Graphs B and D are not.
    • A: the four inputs $-3, -1, 2, 4$ are all different. (The repeated output $3$ is fine.)
    • B: not a function — the vertical line $x = 1$ passes through $(1,2)$ and $(1,-3)$.
    • C: the four inputs $-4, -2, 0, 2$ are all different; every output is $2$, which is allowed.
    • D: not a function — the vertical line $x = 0$ passes through $(0,0)$ and $(0,3)$.
29. Yes. If every $x$-value is different, then no input appears twice, so no input can have two different outputs. That is the whole requirement.
30. The rule constrains inputs, not outputs: it asks that each input have exactly one output, and says nothing about how many inputs may share an output. So $\{(2,8),(3,8)\}$ is a function — asked for the output at $2$ there is one answer, and asked at $3$ there is one answer. A repeated input can break the rule because it can supply two answers to the same question: in $\{(4,1),(4,6)\}$, asked for the output at $4$, the relation says both $1$ and $6$. Note that a repeated input only breaks the rule when the outputs differ — $\{(4,1),(4,1)\}$ is still a function.
31. Function. Each of the inputs $1, 2, 3, 4$ appears once, so every number of walks has exactly one price. If the list were not a function, some number of walks would carry two different prices, and a customer asking "what does three walks cost?" would get two answers with no way to know which one to pay.
32. The student applied the rule to the outputs instead of the inputs. The rule limits how many outputs one input may have; it places no limit on how often an output is reused. Here the inputs $2$, $4$, and $6$ each appear once, so each has exactly one output. Correct verdict: it is a function.

Exit ticket 7.2

33. Yes. The inputs $5$, $6$, $7$ are all different, so each has exactly one output.
34. No. The input $0$ has outputs $4$ and $6$.
35. Yes. The pair $(3,9)$ is listed twice, so the input $3$ has the single output $9$, and the input $4$ has the single output $16$.
36. Vertical line test: if any vertical line passes through two or more of the plotted points, the relation is not a function; if every vertical line passes through at most one plotted point, it is. Two points on one vertical line have the same $x$-coordinate and different $y$-coordinates, so the line detects exactly the forbidden situation — one input with two different outputs.

Lesson 7.3 — Domain and Range

Guided practice

37. Domain $\{1,2,3\}$; range $\{4,5,6\}$
38. Domain $\{-3,-1,4\}$; range $\{0,2\}$. The output $2$ occurs twice and is listed once.
39. Domain $\{2,5\}$; range $\{7,9\}$. The pair $(2,7)$ is listed twice, so neither $2$ nor $7$ is written twice.
40. Domain $\{0,1,2,3\}$; range $\{-1,3,5\}$. The output $-1$ fills two rows and is listed once.
41. Domain $\{-2,0,1,3\}$; range $\{1,3,4\}$
42. A domain and a range are sets, and a set records which values occur, not how many times. Writing $1$ twice in the range would not add any information — the range already says that $1$ is an output — so each value is listed once.
43. Graph A: domain $\{-3,-1,2,4\}$; range $\{0,1,3\}$. The output $3$ occurs at $x = -1$ and at $x = 2$, so it is listed once.
44. Graph C: domain $\{-4,-2,0,2\}$; range $\{2\}$. All four dots share the output $2$.

Independent practice

45. a) Domain $\{-5,-4,0,3\}$; range $\{2,4,6,8\}$ b) Domain $\{1,2,3\}$; range $\{-3\}$ c) Domain $\{-2,0,5\}$; range $\{1,7,9\}$ — the pair $(-2,9)$ is listed twice, and the range is written in increasing order d) Domain $\{0,2,4,6\}$; range $\{0,1,2,3\}$ — the inputs were listed in decreasing order in the table, so they have to be re-sorted
46. Domain $\{5,10,15,20\}$; range $\{2.5,5,7.5,10\}$
47. Graph B: domain $\{-2,1,3\}$; range $\{-3,-1,2,4\}$. The input $1$ appears twice, so the domain lists it once. (Graph B is not a function, which is why its range happens to be larger than its domain.)
48. Graph D: domain $\{0,2,4\}$; range $\{-2,0,1,3\}$
49. $\{(1,2),(2,4),(3,6),(4,8)\}$; range $\{2,4,6,8\}$
50. $\{(-1,5),(0,5),(1,5)\}$. It is a function: each of the three inputs has exactly one output. Every input must be paired with something, and $5$ is the only output available.
51. Yes. Several inputs may share one output, which shrinks the range without shrinking the domain. Example: $\{(1,-3),(2,-3),(3,-3)\}$ has a domain of three values and a range of one.
52. No. Each input of a function contributes exactly one output, so the outputs can be no more numerous than the inputs. Any attempt to give a fourth output to a three-input function would require some input to have two outputs, which is precisely what a function forbids.
53. Domain $\{0,1,2,3\}$; range $\{35,40,45,50\}$. The domain is the numbers of minutes at which the student's progress was recorded; the range is the numbers of pages left at those times.
54. The student copied the output column instead of forming a set, so the repeated $3$ was written twice. A range lists each value once. Correct range: $\{3,5\}$.

Exit ticket 7.3

55. Domain $\{-4,0,2,6\}$; range $\{1,3,7\}$. The output $3$ occurs twice and is listed once.
56. Domain $\{2,4,6,8\}$; range $\{-6,-4,-2,0\}$. Both sets are re-sorted into increasing order, since the table listed them descending.
57. Domain $\{3,9\}$; range $\{1,3\}$. The pair $(3,3)$ is listed twice.
58. No repeats, because a domain and a range are sets, which record which values occur rather than how many times — listing a value twice adds nothing. Increasing order, because it is a convention that makes two answers directly comparable and makes a missing value easy to notice; the order does not change the set itself.

Lesson 7.4 — Putting It All Together

Guided practice

59. $\{(1,4),(2,8),(3,12),(4,16),(5,20)\}$
60. Yes. Each of the inputs $1, 2, 3, 4, 5$ appears exactly once, so every number of bracelets has exactly one cost.
61. Domain $\{1,2,3,4,5\}$; range $\{4,8,12,16,20\}$
62. Three bracelets cost $12.
63. Not a function. The input $2$ appears twice, once with the output $7$ and once with the output $9$.
64. Function. Domain $\{-1,0,1,2\}$; range $\{-1,0,1,2\}$.

Independent practice

65. a) Function; domain $\{2,4,6,8\}$, range $\{1,2,3,4\}$ b) Not a function — the input $-3$ has outputs $5$ and $-5$; domain $\{-3,0\}$, range $\{-5,0,5\}$ c) Function — the pair $(1,7)$ is listed twice, so the input $1$ has the single output $7$; domain $\{1,2,3\}$, range $\{7,9\}$
66. Function. The inputs $1, 2, 3, 4, 5$ each appear once; the outputs $62$ and $65$ repeat, which is allowed. Domain $\{1,2,3,4,5\}$; range $\{62,65,68\}$.
67. The vertical line $x = 0$, which passes through $(0,0)$ and $(0,3)$.
68. Function; domain $\{1,2,3,4\}$, range $\{5,7,9,11\}$. The pair $(4,11)$ means a $4$-mile ride costs $11 — the $3 pickup fee plus $4 \times \$2 = \$8$.
69. $\{(-2,4),(-1,1),(0,0),(1,1),(2,4)\}$. It is a function: each of the five inputs appears once. Range $\{0,1,4\}$ — the outputs $1$ and $4$ each occur twice and are listed once.
70. Any relation in which one input carries two different outputs, repaired by removing that conflict. For example $\{(1,5),(3,2),(3,7),(6,0)\}$ is not a function because the input $3$ has outputs $2$ and $7$. Changing the second $3$ to a $4$ gives $\{(1,5),(3,2),(4,7),(6,0)\}$, in which every input appears exactly once, so each has exactly one output. (Changing the $7$ to a $2$ also works: the input $3$ would then be listed twice with the same output $2$.)
71. Every point on one vertical line has the same $x$-coordinate, which is the same input. Two different points on that line therefore share an input and differ in their $y$-coordinate, which is the output. So "a vertical line meets two plotted points" and "one input is paired with two different outputs" describe the same situation — one is the picture and the other is the sentence.
72. Repetition of an input is not what breaks the rule; two different outputs for one input is. Here the input $1$ is paired with $2$ both times, so asked for the output at $1$ the relation gives one answer, and the two listings plot as a single point. Correct verdict: it is a function.
73. The student listed the outputs. The domain is the set of first coordinates, and the range is the set of second coordinates. Correct domain $\{2,4\}$ (re-sorted into increasing order); correct range $\{1,3\}$.
74. Function. The inputs $1, 2, 3, 4$ each appear once. Domain $\{1,2,3,4\}$; range $\{0,0.5,1.5\}$ — the output $1.5$ occurs on two days and is listed once, and the values are written in increasing order.

Exit ticket 7.4

75. Function — the inputs $0, 3, 6, 9$ are all different. Domain $\{0,3,6,9\}$; range $\{2,4,8\}$, since the output $2$ occurs twice.
76. Not a function — the input $-2$ has outputs $1$ and $5$. Domain $\{-2,0,2\}$; range $\{1,5,7\}$, since the output $5$ occurs twice.
77. Five bracelets cost $20.
78. Read the ordered pairs off whichever representation you are given; decide whether it is a function by looking for a single input paired with two different outputs; then list the inputs as the domain and the outputs as the range, each value once and in increasing order.

Chapter 7 Review

Part A — Determining whether a relation is a function (8.PFA.2a)

79. Yes. The inputs $1$, $2$, $3$ are all different, so each has exactly one output.
80. No. The input $-5$ has outputs $2$ and $3$.
81. Yes. The inputs $0, 1, 2, 3$ each appear once; the output $6$ repeating is irrelevant to the rule.
82. Yes. The pair $(8,2)$ is listed twice, so the input $8$ has the single output $2$.
83. Function. The inputs $1, 3, 5, 7$ are all different. The outputs $2$ and $4$ each repeat, which is allowed.
84. Not a function. The input $4$ appears twice, with outputs $3$ and $5$.
85. Yes, Graph A is a function. Its inputs $-3, -1, 2, 4$ are all different, so no vertical line meets two dots.
86. No. The vertical line $x = 1$ passes through $(1,2)$ and $(1,-3)$.
87. A relation is a function when each input is paired with exactly one output — ask "what is the output at this input?" and there must be exactly one answer for every input. A repeated ordered pair does not break this, because the repeat gives the same answer again: in $\{(8,2),(8,2),(9,4)\}$ the input $8$ still has the single output $2$, and on a graph the two copies plot as one point. What breaks the rule is one input with two different outputs.
88. Yes, it can — but only if every repetition is a repetition of the whole pair. Ten listings with six different $x$-values means four listings repeat an $x$-value already used. If each of those repeats carries the same output as the earlier listing of that input, then every input still has exactly one output and the relation is a function. If even one repeated input carries a different output, the relation is not a function.

Part B — Identifying the domain and range (8.PFA.2b)

89. Domain $\{-2,1,3\}$; range $\{4,5,6\}$
90. Domain $\{0,2,4\}$; range $\{-2\}$
91. Domain $\{5,7\}$; range $\{1,3\}$ — the pair $(5,1)$ is listed twice
92. Domain $\{10,20,30\}$; range $\{0.5,1,1.5\}$
93. Domain $\{-2,0,1,3\}$; range $\{1,3,4\}$
94. Graph C: domain $\{-4,-2,0,2\}$; range $\{2\}$
95. Graph D: domain $\{0,2,4\}$; range $\{-2,0,1,3\}$
96. $\{(-3,-4),(0,-1),(3,2)\}$; range $\{-4,-1,2\}$
97. Any function with those four inputs and exactly two distinct outputs. For example $\{(1,0),(2,0),(3,5),(4,5)\}$, with range $\{0,5\}$. Each input appears once, so it is a function, and two inputs sharing an output is allowed.
98. Each input of a function contributes exactly one output, so the outputs are at most as numerous as the inputs — and fewer whenever two inputs share an output. Producing more range values than domain values would force some input to supply two different outputs, which is exactly what a function forbids.

Part C — Mixed application and reasoning

99. Function; each of the inputs $1$ through $5$ appears once. Domain $\{1,2,3,4,5\}$; range $\{3,6,9,12,15\}$. The pair $(4,12)$ means $4$ cups of flour make $12$ cookies.

100. Function; domain $\{1,2,3,4,5\}$, range $\{4,8,12,16,20\}$. No — adding $(3,15)$ would give the input $3$ two different outputs, $12$ and $15$, so the relation would stop being a function. In the situation, a customer asking the price of three bracelets would be told both $12 and $15.

101. The student tested whether the points were different rather than whether an input was repeated with two outputs. The points $(2,1)$ and $(2,5)$ are indeed different points, and that is the problem: they share the input $2$ and give it two outputs, so the vertical line $x = 2$ meets two dots. Correct verdict: it is not a function.

102. Every point on a single vertical line shares one $x$-coordinate, that is, one input. So a vertical line meeting two or more plotted points shows one input paired with two or more different outputs, which is exactly the definition being violated. If no vertical line ever meets two points, then no input was ever used twice with different outputs, and the definition is satisfied.

103. Any table with those five inputs whose $y$-row contains exactly three distinct values. For example:

     $x$	$1$	$2$	$3$	$4$	$5$
     $y$	$4$	$4$	$7$	$7$	$9$

     Range $\{4,7,9\}$. It is a function because each input appears exactly once; two pairs of inputs sharing an output is allowed.

104. $\{(1,12),(2,21),(3,30),(4,39)\}$, since the cost is $\$3$ plus $\$9$ per pizza: $3 + 9 = 12$, $3 + 18 = 21$, $3 + 27 = 30$, $3 + 36 = 39$. It is a function — each of the inputs $1, 2, 3, 4$ appears once. Domain $\{1,2,3,4\}$; range $\{12,21,30,39\}$.

Workbook-only items

Page 2, fill in the blanks. A relation is any set of ordered pairs. In $(x, y)$ the first number is the input and the second is the output. A relation in this chapter has at most 10 ordered pairs. Discrete means separated, countable with gaps between the values, so the graph is a set of separate dots and never a line.

Page 6, complete the rule. A function is a relation in which each input is paired with exactly one output. A repeated output never breaks the rule. A repeated input breaks the rule only when the two outputs are different.

Page 6, repeated-pair frame. For $\{(1,3),(2,5),(1,3),(4,6)\}$: the input $1$ appears 2 times, its output is 3 both times, so the input has one output. Function? Yes.

Page 7, complete the test. If any vertical line passes through two or more plotted points, the relation is not a function.

Page 8, item 28 table.

Page 11, complete the definitions. The domain is the set of all inputs — the $x$-values. The range is the set of all outputs — the $y$-values. Two rules: no repeats, and increasing order.

Page 16, the routine. Step 1: what are the ordered pairs? Step 2: is it a function — look for one input with two different outputs? Step 3: what are the domain and the range?

Page 18, item 70 frame. Any not-a-function relation with four pairs, repaired by one change. Sample: $\{(1,5),(3,2),(3,7),(6,0)\}$ becomes $\{(1,5),(3,2),(4,7),(6,0)\}$. The change works because the input $3$ no longer carries two different outputs; every input now appears exactly once.

Page 23, item 103 table. Sample completion of the $y$-row: $4$, $4$, $7$, $7$, $9$, giving range $\{4,7,9\}$. Any $y$-row with exactly three distinct values is correct.

Blank-grid pages. Wherever students plot a relation, the expected work is dots only — one dot per ordered pair, never a connecting segment. A response that connects the dots should be corrected even if all the dots are placed correctly, because the segments assert outputs the relation never listed.