1. (partially visible) a function? lesson 6-1 8.f.1.1
2. can an arrow or arrows be drawn from 10.3 so the relation in the diagram is a function? explain your answer. lesson 6-1 8.f.1.1
diagram: domain (4.7, 7.2, 9.6, 10.3) to range (5, 7, 10) with existing arrows
3. keon recorded the amount of water used per load in different types of washing machines. lesson 6-1 8.f.1.1
a. is the relation a function?
b. what are the domain and range of the relation?
chart: type of washing machine (older top loading, new standard model, energy efficient); age of washing machine (years) (6, 4, 2); gallons of water (42, 28, 14)
4. diego took 3 math tests this year. the number of hours he spent studying for each test and the corresponding grades he earned are shown in the table. is the relation of hours of study time to grade earned on a test a function? explain why. use the graph to justify your answer. lesson 6-2 8.f.1.1
table: hours (4, 6, 6); grade (75, 75, 82)
graph: x-axis study time (hours) 0-9, y-axis grade earned 0-90, with axes labeled
5. is the function shown linear or nonlinear? explain your answer. lesson 6-3 8.f.1.2
graph: x-axis 0-9, y-axis 0-9, curve starting at (0,8), peaking, then falling to (8,0)

Question

1. (partially visible) a function? lesson 6-1 8.f.1.1
2. can an arrow or arrows be drawn from 10.3 so the relation in the diagram is a function? explain your answer. lesson 6-1 8.f.1.1
diagram: domain (4.7, 7.2, 9.6, 10.3) to range (5, 7, 10) with existing arrows
3. keon recorded the amount of water used per load in different types of washing machines. lesson 6-1 8.f.1.1
   a. is the relation a function?
   b. what are the domain and range of the relation?
chart: type of washing machine (older top loading, new standard model, energy efficient); age of washing machine (years) (6, 4, 2); gallons of water (42, 28, 14)
4. diego took 3 math tests this year. the number of hours he spent studying for each test and the corresponding grades he earned are shown in the table. is the relation of hours of study time to grade earned on a test a function? explain why. use the graph to justify your answer. lesson 6-2 8.f.1.1
table: hours (4, 6, 6); grade (75, 75, 82)
graph: x-axis study time (hours) 0-9, y-axis grade earned 0-90, with axes labeled
5. is the function shown linear or nonlinear? explain your answer. lesson 6-3 8.f.1.2
graph: x-axis 0-9, y-axis 0-9, curve starting at (0,8), peaking, then falling to (8,0)

Accepted Answer

### Question 2
# Brief Explanations:
To determine if a relation is a function, each element in the domain must map to exactly one element in the range. In the given diagram, the domain element $ 10.3 $ currently has no arrow. If we draw an arrow from $ 10.3 $ to any element in the range (5, 7, or 10), we need to check if it violates the function definition. Since adding an arrow from $ 10.3 $ to one element in the range will mean each domain element (4.7, 7.2, 9.6, 10.3) maps to one range element (assuming we pick one for 10.3), it can be a function. The key is that each domain element has at most one range element. So, yes, an arrow can be drawn (e.g., to 5, 7, or 10) and the relation will still be a function because each domain element will have a unique mapping (or we assign one mapping to 10.3).
# Answer:
Yes, an arrow can be drawn from $ 10.3 $ to make the relation a function. This is because a function requires each element in the domain to map to exactly one element in the range. By drawing an arrow from $ 10.3 $ to one of the range elements (5, 7, or 10), each domain element (4.7, 7.2, 9.6, 10.3) will map to exactly one range element, satisfying the function definition.

### Question 3a
# Brief Explanations:
A relation is a function if each input (in the domain) has exactly one output (in the range). Here, the "Age of Washing Machine (years)" column has values 6, 4, 2 (each unique), and each age maps to one "Gallons of Water" value (42, 28, 14 respectively). So each input (age) has one output (gallons), so it's a function.
# Answer:
Yes, the relation is a function. This is because each age of the washing machine (input) is associated with exactly one amount of water used (output), satisfying the definition of a function (each domain element has exactly one range element).

### Question 3b
# Brief Explanations:
The domain is the set of all input values (age of washing machine), which are 6, 4, 2. The range is the set of all output values (gallons of water), which are 42, 28, 14. We list them in set notation, ensuring uniqueness (though they are already unique here).
# Answer:
- Domain: $ \{2, 4, 6\} $ (or $ \{6, 4, 2\} $, order doesn't matter in a set)
- Range: $ \{14, 28, 42\} $ (or $ \{42, 28, 14\} $, order doesn't matter in a set)

### Question 4
# Brief Explanations:
A relation is a function if each input (hours studied) has exactly one output (grade earned). Looking at the table, when hours = 4, grade = 75; when hours = 6, grade = 75 and 82? Wait, no—wait, the table: hours are 4, 6, 6; grades are 75, 75, 82. Wait, the input "6" (hours) has two outputs: 75 and 82? Wait, no, let's check again. Wait, the table: first row (hours): 4, 6, 6; second row (grades): 75, 75, 82. Wait, actually, the first "6" hours maps to 75, the second "6" hours maps to 82? Wait, no, maybe it's a typo, but in a function, each input must have one output. Wait, but maybe the table is: hours 4 → 75, hours 6 → 75, hours 6 → 82? No, that can't be. Wait, no—maybe the table is three entries: (4,75), (6,75), (6,82). So the input 6 has two outputs: 75 and 82. But wait, the question is "Is the relation of hours of study time to grade earned on a test a function?" Wait, but maybe I misread. Wait, the table: hours: 4, 6, 6; grades: 75, 75, 82. Wait, no—maybe the second 6 is a different case? No, in a function, each input (x - value, hours) must have exactly one output (y - value, grade). So if x = 6 has two y - values (75 and 82), then it's not a function? Wait, but wait, maybe the table is (4,75), (6,75), (6,82) – no, that would mean 6 maps to two grades, so not a function. But wait, the graph: let's check the graph. The graph has study time (x) and grade (y). If we plot (4,75), (6,75), (6,82), then the vertical line x = 6 would intersect two points, so it's not a function. But wait, maybe the table is (4,75), (6,75), (6,82) – no, that's two outputs for x = 6. Wait, but maybe I made a mistake. Wait, the question says "Is the relation of hours of study time to grade earned on a test a function? Explain why. Use the graph to justify your answer." So, in the table, hours (x) are 4, 6, 6; grades (y) are 75, 75, 82. So x = 6 is paired with y = 75 and y = 82. In a function, each x must have only one y. So the relation is not a function? Wait, no—wait, maybe the table is (4,75), (6,75), (6,82) – no, that's two y's for x = 6. But the graph: if we plot these points, (4,75), (6,75), (6,82), then the vertical line x = 6 will pass through two points, violating the vertical line test (a test for functions: a graph represents a function if no vertical line intersects the graph more than once). So the relation is not a function because the input (hours = 6) has two different outputs (grades = 75 and 82), and the graph would fail the vertical line test (a vertical line at x = 6 would intersect two points). Wait, but wait—maybe the table is (4,75), (6,75), (6,82) – no, that's two outputs for x = 6. So the answer is no, it's not a function because one input (6 hours) is associated with two different outputs (75 and 82 grades), and the graph would show two points with the same x - value (6) and different y - values, failing the vertical line test.
Wait, but maybe I misread the table. Let me check again. The table:

Hours: 4, 6, 6

Grades: 75, 75, 82

So three ordered pairs: (4,75), (6,75), (6,82). So x = 6 has two y - values. Therefore, the relation is not a function because a function requires that each input (x - value) has exactly one output (y - value). The vertical line test: if we draw a vertical line at x = 6, it will intersect the graph at (6,75) and (6,82), so it fails the vertical line test, confirming it's not a function.
# Answer:
No, the relation is not a function. This is because the input (hours of study time) of 6 hours is associated with two different outputs (grades of 75 and 82). When we apply the vertical line test to the graph of these points, a vertical line drawn at $ x = 6 $ (study time of 6 hours) will intersect the graph at two points ($ (6, 75) $ and $ (6, 82) $), which violates the vertical line test (a requirement for a relation to be a function, as a vertical line must intersect the graph at most once for each $ x $-value).

### Question 5
# Brief Explanations:
A linear function has a constant rate of change and its graph is a straight line. The given graph is a curve (it has a peak and then decreases, forming a parabola - like or curved shape), not a straight line. So it's nonlinear because the graph is not a straight line (and the rate of change is not constant).
# Answer:
The function is nonlinear. This is because the graph of the function is a curve (not a straight line). A linear function has a constant rate of change and its graph is a straight line, while this graph does not satisfy that (it has a curved shape with a changing rate of change), so it is nonlinear.

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