Question

Think about the three representations of functions you learned above. Create a table, map, and graph of a function and a non-function (one type for each). After you have created and shared your own representations, identify the relationships shared by two of your peers.

In your response to each post:

Identify the type of representation.
Identify if it is a function or non-function.
Post a detailed response to the relations and functions prompt. Then respond to two classmates' posts.

Answer 1

Here's a table, map, and graph representing a function and a non-function:

### Function:
**Function Representation:**
1. **Table:**

| x | f(x) |
|---|------|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
| 5 | 10 |

2. **Mapping:**

- 1 → 2
- 2 → 4
- 3 → 6
- 4 → 8
- 5 → 10

3. **Graph:**

- **Graph Description:**
The graph of the function can be plotted as points on a Cartesian coordinate system:

- (1, 2)
- (2, 4)
- (3, 6)
- (4, 8)
- (5, 10)

### Non-Function:
**Non-Function Representation:**
1. **Table:**

| x | g(x) |
|---|------|
| 1 | 2 |
| 1 | 3 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |

2. **Mapping:**

- 1 → 2
- 1 → 3
- 2 → 4
- 3 → 6
- 4 → 8

3. **Graph:**

- **Graph Description:**
The graph of this non-function will show that the point (1, 2) and (1, 3) overlap vertically on the y-axis at x = 1, illustrating that for x = 1, there are two different outputs.

### Function vs Non-Function Analysis:

#### Function:
For the function \(f(x)\), each input value \(x\) corresponds to exactly one output value \(f(x)\). This means that for every unique \(x\), there is a single unique \(f(x)\). This is a crucial characteristic of functions and ensures that we can predict the output accurately from a given input.

#### Non-Function:
In contrast, the non-function \(g(x)\) demonstrates that the value \(x=1\) corresponds to two different outputs (2 and 3). This violates the definition of a function, as an input must map to exactly one output, not multiple outputs. Therefore, \(g(x)\) is not a function because it fails to satisfy the vertical line test, which states that if a vertical line intersects the graph of the relation more than once, it is not a function.

### Responses to Classmates:
Once your classmates post their examples, you can evaluate them based on the criteria above, noting the type of representation used, whether their relation is a function or non-function, and engaging in a discussion about the implications of their findings.