Appendix A — Answer Key, Chapter 4: Functions, Domain, and Range

SOL A.F.2 (a) · Covers textbook Chapter 4 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 116 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Conventions used in every answer below: a domain or a range of a relation with finitely many pairs is written inside braces, with each value listed once and in increasing order. For a continuous graph, both sets are described in words; interval notation is never required. 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 relations used repeatedly in the chapter, for reference:

• Mapping M $= \{(1,2),(4,2),(7,2)\}$, Mapping N $= \{(-1,5),(-1,8),(3,0)\}$, Mapping P $= \{(0,0),(2,4),(5,25)\}$
• Graph A $= \{(-4,2),(-2,0),(1,3),(3,3)\}$ and Graph B $= \{(-3,1),(-3,-2),(0,4),(2,-1)\}$ are discrete; Graph C is the line $y = -x + 2$; Graph D is the circle of radius $3$ centered at the origin
• The discrete graph of Lesson 4.4 $= \{(-4,1),(-1,3),(2,-2),(4,3)\}$
• Tickets $= \{(1,8),(2,16),(3,24),(4,32),(5,40),(6,48)\}$

Lesson 4.1 — Relations and Their Four Representations

Guided practice

1. $x$	$y$
   $-3$	$4$
   $0$	$1$
   $2$	$-5$

2. $\{(-2,0),(1,3),(4,6),(7,9)\}$
3. Input $-6$, output $11$.
4. $\{(1,2),(4,2),(7,2)\}$
5. A set of ordered pairs, a table, a mapping, and a graph.
6. A discrete graph is a set of separated points; it states a fact at each plotted input and says nothing between them. A continuous graph is an unbroken curve or line; every input in the stretch it covers has a point of the graph above or below it.

Independent practice

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

8. $x$	$y$
   $-4$	$-4$
   $-2$	$0$
   $1$	$3$
   $5$	$-1$

9. $\{(-3,2),(-1,0),(2,3),(4,-1)\}$
10. The left box holds the inputs $2$, $3$, $6$; the right box holds the outputs $1$ and $5$, with $5$ written once even though it is used twice. Three arrows are drawn, one per ordered pair: $2 \rightarrow 5$, $3 \rightarrow 5$, $6 \rightarrow 1$.
11. $\{(-1,-5),(0,-3),(2,1),(5,7)\}$, from $2(-1) - 3 = -5$, $2(0) - 3 = -3$, $2(2) - 3 = 1$, and $2(5) - 3 = 7$.

12. $x$	$y$
    $-1$	$-5$
    $0$	$-3$
    $2$	$1$
    $5$	$7$

13. The table makes a repeated input easiest to spot, because every input sits in one column where a repeat is directly visible. The mapping makes it easiest to miss in one specific way: the repeated input is written only once in the left box, so the repetition shows up only as a second arrow leaving that value, which is easy to overlook. A graph hides a repeated identical pair completely, since the two points land on top of each other, though it displays a repeated input with two different outputs very clearly as two points on one vertical line.
14. $\{(1,6),(2,12),(3,18),(4,24)\}$. The input is the number of hours the booth is rented; the output is the cost in dollars.
15. The student reversed the coordinates, writing each output first. A table pairs each $x$ with the $y$ beside it, and in an ordered pair the input is always written first. Correct relation: $\{(3,-1),(8,5)\}$.
16. A relation is a set, and a set is determined by which members it contains, not by the order they are listed in — so $\{(1,2),(2,1)\}$ and $\{(2,1),(1,2)\}$ contain exactly the same two pairs and are the same relation. Within a single pair, order does carry meaning: $(1,2)$ says the input $1$ has output $2$, while $(2,1)$ says the input $2$ has output $1$. Those are different statements and different plotted points.

Exit ticket 4.1

17. $x$	$y$
    $-2$	$7$
    $0$	$7$
    $5$	$-3$

18. $\{(1,-4),(2,-8),(3,-12)\}$
19. Input $9$, output $-2$.
20. A set of ordered pairs — read each pair, input first. A table — read across each row, the input column giving the first coordinate. A mapping — follow each arrow from a value in the left box to a value in the right box. A graph — read each point's coordinates, across first and then up or down.

Lesson 4.2 — Deciding Whether a Relation Is a Function

Guided practice

21. Yes. The inputs $-3$, $0$, and $4$ are all different, so no input can carry two outputs.
22. No. The input $1$ is paired with $6$ and also with $9$, which is two different outputs for one input.
23. Yes. Each of the inputs $2$, $5$, $8$ appears once, so each has exactly one output; the repeated output $7$ is allowed.
24. Yes. The input $-1$ appears twice, but its output is $3$ both times, so the input $-1$ still has exactly one output.
25. The mapping on the left is the function: exactly one arrow leaves each of the inputs $-2$, $0$, and $3$. On the right, two arrows leave the input $2$, sending it to both $5$ and $9$, so that relation is not a function.
26. Yes, it is a function. Two arrows arrive at the output $4$, which the definition allows, but only one arrow leaves each of the inputs $1$, $2$, and $3$, which is the whole requirement.

Independent practice

27. a) Function. The inputs $-5$, $-3$, $0$, $4$ are all different; the output $1$ repeating does not matter. b) Not a function. The input $6$ has outputs $2$ and $8$. c) Function. The inputs $0$, $1$, $4$, $9$ each appear once. d) Function. The pair $(-2,5)$ is listed twice, so the input $-2$ has the single output $5$.
28. Function. The inputs $2, 4, 6, 8, 10$ each appear once. The outputs $12$ and $15$ each occur twice, which the rule allows.
29. Not a function. The input $-4$ fills two rows, once with the output $6$ and once with the output $9$. (The repeated output $6$ at $x = -4$ and $x = -2$ is not the problem.)
30. Function. Every row is the identical pair $(5,2)$, so the input $5$ has exactly one output, $2$. Domain $\{5\}$; range $\{2\}$.
31. Mappings M and P are functions; Mapping N is not. In N, the input $-1$ has two arrows leaving it, one to $5$ and one to $8$. (In M, three arrows arrive at the single output $2$, which is allowed.)
32. Yes, but only if every repetition repeats the whole pair. Six listings with four different inputs means two listings reuse an input already present; if each of those carries the same output as the earlier listing of that input, every input still has exactly one output and the relation is a function. If even one repeated input carries a different output, it is not.
33. The definition constrains inputs, not outputs: it asks that each input have exactly one output and says nothing about how many inputs may share one. So $\{(2,7),(5,7)\}$ is a function — asked for the output at $2$ there is one answer, and asked at $5$ there is one answer. A repeated input can break the rule because it can supply two answers to a single question: in $\{(1,6),(1,9)\}$, asked for the output at $1$, the relation says both $6$ and $9$. A repeated input breaks the rule only when the outputs differ; $\{(1,6),(1,6)\}$ is still a function.
34. Function. The inputs $214$, $215$, $216$ each appear once, so each student ID has exactly one locker. The pair $(216,5)$ says student $216$ is assigned locker $5$ — two students sharing locker $5$ is a repeated output, which the definition permits and which the school may well intend. Containing both $(214,5)$ and $(214,9)$ would give one student two different lockers, so the relation would not be a function and the question "where does student $214$ go?" would have two answers.
35. The student applied the rule to the outputs instead of the inputs. The rule limits how many outputs one input may have and places no limit on how often an output is reused. Here the inputs $3$, $5$, $9$ each appear once, so each has exactly one output. Correct verdict: it is a function.
36. $f(4) = 2$ says that the function named $f$ pairs the input $4$ with the output $2$ — the same fact as the ordered pair $(4,2)$, which is in the set. The relation $\{(1,6),(1,9)\}$ cannot be named this way because $f(1)$ would have to equal both $6$ and $9$, and the notation $f(1)$ can only stand for one number. The notation presumes exactly what a function guarantees.

Exit ticket 4.2

37. Yes. The inputs $7$, $8$, $9$ are all different, so each has exactly one output.
38. No. The input $0$ has outputs $5$ and $9$.
39. Yes. The pair $(-4,2)$ is listed twice, so the input $-4$ has the single output $2$, and the input $6$ has the single output $2$. A shared output is allowed.
40. In $f(x)$, the letter $f$ names the rule — the whole pairing of inputs with outputs — and the entire symbol $f(x)$ names the output the rule produces at the input $x$. It is not multiplication. Only a function can be written this way, because $f(x)$ has to stand for exactly one number; a relation with two different outputs at some input gives that symbol two values at once.

Lesson 4.3 — Graphs and the Vertical Line Test

Guided practice

41. Yes, the graph on the left is a function, by the vertical line test: every vertical line drawn anywhere on the picture crosses the line exactly once.
42. No, the circle is not a function. The vertical line $x = 2$ passes through about $(2, 3.46)$ and about $(2, -3.46)$, so the input $2$ has two different outputs. (Any vertical line strictly between $x = -4$ and $x = 4$ would do.)
43. Yes, Graph A is a function. Its ordered pairs are $\{(-4,2),(-2,0),(1,3),(3,3)\}$, and the four inputs are all different, so no vertical line meets two points.
44. No. The vertical line $x = -3$ passes through $(-3,1)$ and $(-3,-2)$.
45. Yes. Graph C is the unbroken line $y = -x + 2$, and every vertical line crosses it exactly once.
46. No. Graph D is a circle; the vertical line $x = 0$ passes through $(0,3)$ and $(0,-3)$, so the input $0$ has two different outputs.

Independent practice

47. Function. The inputs $-5$, $-2$, $0$, $3$ are all different, so no vertical line meets two of the points. The output $1$ occurring at both $-2$ and $0$ is allowed.
48. Not a function. The vertical line $x = 1$ passes through $(1,2)$ and $(1,-2)$.
49. No. Every point of the graph of $x = -3$ has the input $-3$, so that single input is paired with infinitely many outputs — and the vertical line $x = -3$ lies along the whole graph, meeting it in more than one point.
50. Yes. A vertical line drawn anywhere crosses $y = 4$ at exactly one point, so every input has exactly one output. Every input shares the output $4$, and a repeated output never breaks the rule.
51. 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 the graph twice" and "one input is paired with two different outputs" describe the same situation — one is the picture and the other is the sentence.
52. The circle fails — for instance $x = 0$ meets a circle centered at the origin at its top and bottom points. The vertical line fails — the line $x = a$ lies along its own graph, meeting it everywhere. The unbroken slanted line and the horizontal line both pass; every vertical line meets each of them exactly once.
53. Function. Each of the inputs $1$ through $6$ appears once, so every ticket count has exactly one total cost. A "no" answer would mean that some number of tickets carried two different prices, and a customer asking the cost of four tickets would be given two answers with no way to know which to pay.
54. The student read the rule as a restriction on outputs. It is a restriction on inputs only: each input must have exactly one output. On $y = 4$ every input does have exactly one output, namely $4$; outputs are allowed to repeat as often as they like. Correct verdict: $y = 4$ is a function.
55. Yes. Two points sharing a $y$-coordinate sit on the same horizontal line, and the test says nothing about horizontal lines. Graph A is an example: $(1,3)$ and $(3,3)$ share the output $3$, yet the four inputs are all different, so every vertical line still meets at most one point.
56. Each reading pairs one time with one temperature, and the station takes exactly one reading per minute, so each input time is plotted with exactly one output temperature. No two plotted points share a time, so no vertical line can pass through two of them, and the graph passes the test.

Exit ticket 4.3

57. Not a function. The vertical line $x = 0$ passes through $(0,3)$ and $(0,-1)$, so the input $0$ has two different outputs.
58. Vertical line test: if any vertical line passes through two or more points of the graph, the relation is not a function; if every vertical line passes through at most one point, it is. It detects a single input paired with two different outputs, because two points on one vertical line share an $x$-coordinate and differ in their $y$-coordinate.
59. Yes. The line $y = x - 1$ is slanted and unbroken, so a vertical line drawn at any input crosses it exactly once — one output per input.
60. A vertical line drawn strictly inside a circle enters it and leaves it, crossing the curve twice. Those two crossings share an input and have different outputs, so the definition of a function is violated at every such input.

Lesson 4.4 — Domain and Range

Guided practice

61. Domain $\{-3,1,6\}$; range $\{0,4\}$. The output $4$ occurs at two inputs and is listed once.
62. Domain $\{2,5\}$; range $\{-1,9\}$. The pair $(2,9)$ is listed twice, so neither value is written twice; the relation is a function.
63. Domain $\{-6,-2,0,3\}$; range $\{-4,5,7\}$. The output $5$ fills two rows and is listed once, and the range is sorted.
64. Mapping M: domain $\{1,4,7\}$; range $\{2\}$.
65. Domain $\{-4,-1,2,4\}$; range $\{-2,1,3\}$. The output $3$ happens at $x = -1$ and at $x = 4$, so it appears once.
66. Domain: all real numbers. Range: all real numbers. The arrows on both ends say the line never stops, and a slanted line eventually reaches every height.

Independent practice

67. a) Domain $\{-7,-4,2,5\}$; range $\{-6,0,3,8\}$ b) Domain $\{0,3,9\}$; range $\{-5\}$ c) Domain $\{-2,4,6\}$; range $\{1,7\}$ — the pair $(4,1)$ is listed twice, and the output $7$ occurs at two inputs d) Domain $\{4,6,8,10\}$; range $\{2,4,6,8\}$ — the table listed the inputs in decreasing order, so both sets are re-sorted
68. Domain $\{3,6,9,12\}$; range $\{0,0.5,1,1.5\}$. Both sets are re-sorted into increasing order.
69. Mapping N: domain $\{-1,3\}$; range $\{0,5,8\}$. (N is not a function, which is why its range holds more values than its domain.)
70. Mapping P: domain $\{0,2,5\}$; range $\{0,4,25\}$.
71. Graph A: domain $\{-4,-2,1,3\}$; range $\{0,2,3\}$. The output $3$ occurs twice and is listed once.
72. Graph B: domain $\{-3,0,2\}$; range $\{-2,-1,1,4\}$. The input $-3$ appears twice, so the domain lists it once.
73. Graph C: domain all real numbers; range all real numbers. The line $y = -x + 2$ is slanted and unbroken in both directions.
74. Domain: all real numbers from $-2$ up to but not including $3$. Range: all real numbers greater than $-1$ and up to and including $4$. The filled-in point at $(-2,4)$ includes both $-2$ and $4$; the open point at $(3,-1)$ excludes both $3$ and $-1$.
75. $\{(1,8),(2,16),(3,24),(4,32),(5,40),(6,48)\}$. It is a function: each of the inputs $1$ through $6$ appears exactly once. Domain $\{1,2,3,4,5,6\}$; range $\{8,16,24,32,40,48\}$. The pair $(4,32)$ means four tickets cost $\$32$.
76. $\{(1,2.25),(2,4.5),(3,6.75),(4,9)\}$. It is a function: each input appears once. Domain $\{1,2,3,4\}$; range $\{2.25,4.5,6.75,9\}$, in dollars. The domain is not "all real numbers from $1$ to $4$" because a customer cannot buy $2.5$ muffins — the situation admits only whole numbers of muffins, which is why the graph is four separate points rather than a segment.
77. No. Each input of a function contributes exactly one output, so the outputs can be no more numerous than the inputs. Producing more range values than domain values would force some input to supply two different outputs, which is precisely what a function forbids. (The range may certainly contain fewer values, as in item 67b.)
78. The student listed the outputs. The domain is the set of first coordinates and the range is the set of second coordinates. Correct domain $\{1,5\}$, re-sorted into increasing order; correct range $\{2,8\}$.

Exit ticket 4.4

79. Function — the pair $(4,-2)$ is listed twice, so the input $4$ has the single output $-2$. Domain $\{-8,0,4\}$; range $\{-2,3\}$, since the output $3$ occurs at two inputs.
80. Domain $\{1,3,5\}$; range $\{-2,6\}$. The output $-2$ fills two rows and is listed once.
81. Domain $\{-4,-1,2,4\}$; range $\{-2,1,3\}$.
82. On a discrete graph you can list the domain, because there are finitely many points: read the $x$-coordinate of each one, drop repeats, and sort. On an unbroken line there are infinitely many points, so no list is possible and the domain is described in words instead — "all real numbers," or a description bounded by the endpoints, which also has to say whether each endpoint is included.

Chapter 4 Review

Part A — Relations given as a set of ordered pairs

83. Function; the inputs $1$, $2$, $3$ are all different. Domain $\{1,2,3\}$; range $\{4,8,12\}$.
84. Not a function; the input $-2$ has outputs $6$ and $-6$. Domain $\{-2,5\}$; range $\{-6,0,6\}$.
85. Function; the inputs $0$, $4$, $9$ each appear once and the repeated output $7$ is allowed. Domain $\{0,4,9\}$; range $\{7\}$.
86. Function; the pair $(6,1)$ is listed twice, so the input $6$ has the single output $1$. Domain $\{6,8\}$; range $\{1,3\}$.
87. Function; the four inputs are all different. Domain $\{-1,0,1,2\}$; range $\{-1,0,1,2\}$.
88. Not a function; the input $3$ has outputs $5$ and $9$. Domain $\{3,4\}$; range $\{5,9\}$.
89. Any relation in which one input carries two different outputs, repaired by removing that conflict. For example $\{(1,7),(4,2),(4,9),(6,0)\}$ is not a function, because the input $4$ has outputs $2$ and $9$. Changing the second $4$ to a $5$ gives $\{(1,7),(4,2),(5,9),(6,0)\}$, in which every input appears exactly once, so each has exactly one output. (Changing the $9$ to a $2$ also works: the input $4$ would then be listed twice with the same output.)
90. Yes, it can — but only if every one of the three extra listings repeats a pair already present. Eight listings with five different inputs means three listings reuse an input; if each carries the same output as the earlier listing of that input, then every input still has exactly one output. If even one of them carries a different output, the relation is not a function.

Part B — Relations given as a table

91. Function; the inputs $2, 4, 6, 8$ are all different, and one output shared by all four is allowed. Domain $\{2,4,6,8\}$; range $\{5\}$.
92. Function; the input $-3$ appears twice but carries the output $4$ both times. Domain $\{-3,-1,2\}$; range $\{0,4,8\}$.
93. Not a function; the input $1$ appears in two rows, once with the output $7$ and once with the output $3$. Domain $\{0,1,2\}$; range $\{3,5,7,9\}$.
94. Function; the inputs $10, 20, 30, 40$ each appear once. Domain $\{10,20,30,40\}$; range $\{2.5,5,7.5,10\}$.
95. Function; each of the inputs $1, 2, 3, 4$ appears once, so each number of hours carries exactly one amount of pay. Domain $\{1,2,3,4\}$; range $\{15,30,45,60\}$. The pair $(3,45)$ means three hours of work pays $\$45$, which is $\$15$ per hour.
96. Read straight down the input column and keep track of the inputs you have already seen. If you reach the bottom without meeting a repeat, the relation is a function and you can stop. Whenever you do meet a repeat, look across at the two outputs: if they are the same value, the rule is still intact and you keep going; if they differ, you have found one input with two outputs and the relation is not a function.

Part C — Relations given as a mapping

97. Mapping M is a function: exactly one arrow leaves each of $1$, $4$, and $7$. Domain $\{1,4,7\}$; range $\{2\}$. (Three arrows arrive at $2$, which is allowed.)
98. Mapping N is not a function. The input $-1$ has two arrows leaving it, one to $5$ and one to $8$. Domain $\{-1,3\}$; range $\{0,5,8\}$.
99. Mapping P is a function: one arrow leaves each of $0$, $2$, and $5$. Domain $\{0,2,5\}$; range $\{0,4,25\}$.
100. Yes, it is a function. An arrow that leaves an input announces an output for that input, so two arrows leaving one input would give it two outputs and break the rule. An arrow that arrives at an output only records that some input produced that value; several inputs may produce the same one. Here two arrows arrive at $4$ while exactly one leaves each of $1$, $2$, and $3$.
101. The left box holds $-2$, $0$, $3$ and the right box holds only $7$. Three arrows are drawn, one from each input to $7$. It is a function: exactly one arrow leaves each input, and the three inputs sharing an output is allowed.
102. Look only at the left box and count the arrows leaving each value. If every input has exactly one arrow leaving it, the relation is a function; if any input has two or more, it is not. Counting the arrows arriving at an output tells you nothing, because the definition places no limit on how many inputs may share an output — several arrivals is the ordinary picture of a perfectly good function.

Part D — Relations given as a graph

103. Graph A is a function; the inputs $-4$, $-2$, $1$, $3$ are all different, so no vertical line meets two points. Domain $\{-4,-2,1,3\}$; range $\{0,2,3\}$.
104. Graph B is not a function. The vertical line $x = -3$ passes through $(-3,1)$ and $(-3,-2)$. Domain $\{-3,0,2\}$; range $\{-2,-1,1,4\}$.
105. Graph C is a function; it is the unbroken slanted line $y = -x + 2$, which every vertical line crosses exactly once. Domain: all real numbers. Range: all real numbers.
106. Graph D is not a function. The vertical line $x = 0$ passes through $(0,3)$ and $(0,-3)$. (Any vertical line strictly between $x = -3$ and $x = 3$ also works.)
107. Every vertical line drawn on that picture crosses the line $y = 0.5x + 1$ at exactly one point. So every input has exactly one output, which is the definition of a function; there is no input anywhere on the graph where a second output could be found.
108. Domain: all real numbers from $-2$ up to but not including $3$. Range: all real numbers greater than $-1$ and up to and including $4$. The open point says the graph approaches $(3,-1)$ without reaching it, so $3$ is not an input and $-1$ is not an output; had the point been filled in, both would be included.
109. Function; each of the inputs $1$ through $6$ appears exactly once. Domain $\{1,2,3,4,5,6\}$; range $\{8,16,24,32,40,48\}$. The pair $(6,48)$ means six tickets cost $\$48$.
110. Points sharing a vertical line are exactly the points sharing an input. So "some vertical line meets the graph twice" says "some input is paired with two different outputs," which is word for word the failure the definition forbids, and "no vertical line meets the graph twice" says every input has at most one output. The test is not extra information; it is the definition redrawn.

Part E — Mixed application and reasoning

111. $\{(1,9),(2,13),(3,17),(4,21)\}$, from $5 + 4(1) = 9$, $5 + 4(2) = 13$, $5 + 4(3) = 17$, $5 + 4(4) = 21$. It is a function: each of the inputs $1, 2, 3, 4$ appears once. Domain $\{1,2,3,4\}$; range $\{9,13,17,21\}$. The pair $(3,17)$ means a three-hour rental costs $\$17$ — the $\$5$ deposit plus $3 \times \$4 = \$12$.
112. A repeated output never breaks the rule because the rule governs inputs only: it demands one output per input and says nothing about how many inputs may share an output, so $\{(2,7),(5,7),(8,7)\}$ is a function. A repeated input breaks the rule only when the two outputs differ, because only then does the question "what is the output here?" get two answers: $\{(1,6),(1,9)\}$ fails, while $\{(1,6),(1,6)\}$ is a function, since the second listing repeats the first answer rather than contradicting it.
113. Repetition of an input is not what breaks the rule; two different outputs for one input is. Here the input $2$ is paired with $3$ both times, so the question "what is the output at $2$?" has exactly one answer, and the two listings plot as a single point. Correct verdict: it is a function. Domain $\{2\}$; range $\{3\}$.
114. Both checks look for the same thing. A repeat in the input column of the table means two rows share an $x$-value; plotted, those two rows become two points with the same $x$-coordinate, which is to say two points on one vertical line. If their outputs differ, the table check reports a conflict and the vertical line meets two distinct points — both verdicts are "not a function." If their outputs are identical, the table check clears the repeat and the two points coincide, so no vertical line meets two different points — both verdicts are "function." The two procedures inspect the same fact through different windows.
115. Any function with those four inputs and exactly two distinct outputs. For example $\{(1,0),(2,0),(3,6),(4,6)\}$, with range $\{0,6\}$. It is a function because each of the inputs $1$, $2$, $3$, $4$ appears exactly once, so each has exactly one output; two inputs sharing an output is permitted by the definition.
116. $c(5) = 40$ says that the ticket-cost function pairs the input $5$ with the output $40$: five tickets cost $\$40$. The $5$ is an input, a number of tickets; the $40$ is the output, a total cost in dollars. It is the same fact as the ordered pair $(5,40)$.

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. The four representations the standard names are a set of ordered pairs, a table, a mapping, and a graph. A discrete graph is a set of separated points; a continuous graph is an unbroken curve or line.

Page 6, complete the definition. 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, the mapping rule. In a mapping, count the arrows that leave each input, never the arrows that arrive at an output.

Page 7, repeated-pair frame. For $\{(-1,3),(6,0),(-1,3),(2,8)\}$: the input $-1$ appears 2 times, its output is 3 both times, so the input has one output. Function? Yes.

Page 8, function notation frame. In $f(x)$, the letter $f$ names the rule and the whole symbol names the output at the input $x$. Read aloud: "$f$ of $x$." It is not multiplication. The statement $f(4) = 2$ is the same fact as the ordered pair $(4, 2)$.

Page 10, complete the test. If any vertical line passes through two or more points of the graph, the relation is not a function.

Page 10, two special lines. A horizontal line such as $y = 4$ is a function. A vertical line such as $x = -3$ is not a function.

Page 11, item 52 table.

Page 13, 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 for writing either set: no repeats, and increasing order. For a continuous graph, describe both sets in words.

Page 14, endpoint frame. A filled-in point means the endpoint is included. An open point means the endpoint is not included.

Page 18, item 89 frame. Any not-a-function relation with four pairs, repaired by one change. Sample: $\{(1,7),(4,2),(4,9),(6,0)\}$ becomes $\{(1,7),(4,2),(5,9),(6,0)\}$. The change works because the input $4$ no longer carries two different outputs; every input now appears exactly once.

Page 22, item 115 table. Sample completion of the $y$-row: $0$, $0$, $6$, $6$, giving range $\{0,6\}$. Any $y$-row with exactly two distinct values is correct.

Blank-grid pages. Where students plot a discrete relation, the expected work is dots only — one dot per ordered pair, never a connecting segment, because the segments would assert outputs the relation never listed. Where students are asked to draw a continuous graph, the expected work is an unbroken line drawn all the way across the grid, with arrowheads if the relation has no endpoints and a filled-in or open point wherever it does.