Eighth Grade Identifying Functions Practice (teacher made) - Free Printable

Problem Overview:

The task is to determine which of the given relations are functions. A relation is a function if and only if each input (x-value) is paired with exactly one output (y-value). In other words, no x-value can be associated with more than one y-value.

Solution Approach:

1. Understand the Definition of a Function:
- A function is a special type of relation where each element in the domain (input values, x-values) maps to exactly one element in the range (output values, y-values).
- If any x-value appears more than once with different y-values, the relation is not a function.

2. Analyze Each Relation:
- For each given relation, check if any x-value is repeated with different y-values.
- If such a case exists, the relation is not a function.
- If every x-value is unique or repeats with the same y-value, the relation is a function.

3. Check Graphs:
- Use the Vertical Line Test: If any vertical line intersects the graph at more than one point, the relation is not a function. Otherwise, it is a function.

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Detailed Analysis of Each Part:

#### Part 1: Relations as Sets of Ordered Pairs
1. Relation 1: `{(5, 2), (3, 4), (7, 8), (-8, 3), (-11, 3), (3, 4)}`
- Check for repeated x-values:
- The x-value `3` appears twice: `(3, 4)` and `(3, 4)`.
- Both occurrences have the same y-value (`4`).
- Conclusion: This is a function.

2. Relation 2: `{(-5, 8), (10, 5), (-6, 7), (-9, 14), (0, 0)}`
- Check for repeated x-values:
- All x-values are unique: `-5, 10, -6, -9, 0`.
- Conclusion: This is a function.

3. Relation 3: `{(4, 4), (8, 4), (2, 7), (4, 7), (2, 9), (1, 0)}`
- Check for repeated x-values:
- The x-value `4` appears twice: `(4, 4)` and `(4, 7)`.
- These pairs have different y-values (`4` and `7`).
- Conclusion: This is not a function.

4. Relation 4: `{(1, 2), (3, 4), (5, 6), (7, 8), (9, 8), (10, 11)}`
- Check for repeated x-values:
- All x-values are unique: `1, 3, 5, 7, 9, 10`.
- Conclusion: This is a function.

#### Part 2: Relations as Tables
5. Relation 5:
| x | y |
|---|---|
| 3 | 5 |
| 5 | 7 |
| 7 | 9 |
| 9 | 11 |
- Check for repeated x-values:
- All x-values are unique: `3, 5, 7, 9`.
- Conclusion: This is a function.

6. Relation 6:
| x | y |
|---|---|
| 2 | 16 |
| 4 | 18 |
| 3 | 17 |
- Check for repeated x-values:
- All x-values are unique: `2, 4, 3`.
- Conclusion: This is a function.

7. Relation 7:
| x | y |
|---|---|
| 4 | 6 |
| 6 | 6 |
| 8 | 6 |
- Check for repeated x-values:
- All x-values are unique: `4, 6, 8`.
- Conclusion: This is a function.

#### Part 3: Relations as Graphs
8. Graph 8:
- Apply the Vertical Line Test:
- Draw vertical lines across the graph.
- No vertical line intersects the graph at more than one point.
- Conclusion: This is a function.

9. Graph 9:
- Apply the Vertical Line Test:
- Draw vertical lines across the graph.
- One vertical line (e.g., at x = 2) intersects the graph at two points.
- Conclusion: This is not a function.

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Final Answer:

- Relations that are functions: 1, 2, 4, 5, 6, 7, 8
- Relations that are not functions: 3, 9

$$
\boxed{1, 2, 4, 5, 6, 7, 8}
$$