Worksheet for identifying and representing functions, featuring problems with tables, graphs, and ordered pairs.

Problem: Identifying and Representing Functions

The task is to determine whether each given relationship is a function. A function is a special type of relationship where each input (x-value) corresponds to exactly one output (y-value). If any input is associated with more than one output, the relationship is not a function.

Let's solve each part step by step:

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#### 1.
- Diagram: Arrows from inputs (8, 9) to outputs (3, 4, 6, 9).
- Analysis: The input `8` is connected to multiple outputs (`3`, `4`, `6`). Similarly, the input `9` is connected to multiple outputs (`4`, `6`, `9`).
- Conclusion: This is not a function because some inputs are associated with more than one output.

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#### 2.
- Diagram: Arrows from inputs (0, 1, 2, 3) to outputs (15, 14, 13, 12).
- Analysis: Each input is connected to exactly one output.
- Conclusion: This is a function.

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#### 3.
- Diagram: Arrows from inputs (4, 8, 12, 16) to outputs (5, 10, 15).
- Analysis: The input `4` is connected to two outputs (`5` and `10`). Similarly, the input `8` is connected to two outputs (`10` and `15`).
- Conclusion: This is not a function because some inputs are associated with more than one output.

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#### 4.
- Graph: Points plotted on a coordinate plane.
- Analysis: Each x-value (input) appears only once. No x-value is repeated.
- Conclusion: This is a function.

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#### 5.
- Graph: Points plotted on a coordinate plane.
- Analysis: Each x-value (input) appears only once. No x-value is repeated.
- Conclusion: This is a function.

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#### 6.
- Graph: Points plotted on a coordinate plane.
- Analysis: The x-value `2` appears twice, corresponding to two different y-values.
- Conclusion: This is not a function because the same input (`2`) is associated with more than one output.

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#### 7.
- Table:
| Input | Output |
|-------|--------|
| 0 | 4 |
| 1 | 1 |
| 2 | 0 |
| 3 | 4 |
- Analysis: Each input appears only once. No input is repeated.
- Conclusion: This is a function.

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#### 8.
- Table:
| Input | Output |
|-------|--------|
| 1 | 4 |
| 2 | 5 |
| 0 | 6 |
| 1 | 7 |
| 2 | 8 |
- Analysis: The input `1` appears twice, corresponding to two different outputs (`4` and `7`). Similarly, the input `2` appears twice, corresponding to two different outputs (`5` and `8`).
- Conclusion: This is not a function because the same input is associated with more than one output.

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#### 9.
- Set of Ordered Pairs: {(0, 0), (2, 4), (3, 6), (5, 5), (7, 6)}.
- Analysis: Each x-value (input) appears only once. No x-value is repeated.
- Conclusion: This is a function.

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#### 10.
- Set of Ordered Pairs: {(0, 8), (1, 2), (3, 7), (5, 9), (3, 6)}.
- Analysis: The x-value `3` appears twice, corresponding to two different outputs (`7` and `6`).
- Conclusion: This is not a function because the same input is associated with more than one output.

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#### 11.
- Graph: Relationship between hours studied and exam grades.
- Analysis: Each x-value (hours studied) appears only once. No x-value is repeated.
- Conclusion: This is a function.
- Justification: In the context of the graph, the "input" is the number of hours studied, and the "output" is the exam grade earned. Since no student can study the same number of hours and earn two different grades simultaneously, this relationship satisfies the definition of a function.

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#### 12.
- Scenario: Rachel plans to study 2 hours for her next exam.
- Graph: Adding the point (2, ?) to the existing graph.
- Analysis: If Rachel studies 2 hours and earns a grade that is different from any previously recorded grade for 2 hours, it would not affect the conclusion that the relationship is a function. However, if she earns the same grade as someone who already studied 2 hours, the relationship would still be a function because the input (2 hours) would still map to exactly one output (grade).
- Conclusion: Plotting her grade will not change the answer to Exercise 11, as long as the grade she earns does not violate the one-to-one correspondence between inputs and outputs.

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Final Answers:

1. Not a function
2. Function
3. Not a function
4. Function
5. Function
6. Not a function
7. Function
8. Not a function
9. Function
10. Not a function
11. Function (Justified using inputs and outputs in the context of studying hours and grades.)
12. Plotting her grade will not change the answer to Exercise 11, as long as the grade she earns does not violate the one-to-one correspondence between inputs and outputs.

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