Function Example: Table: Representing the function f(x) = 2x using a table: x | f(x) 1 | 2 2 | 4 3 | 6 4 | 8 Map: Representing the function f(x) = 2x using a mapping: Input x: 1, 2, 3, 4 Output f(x): 2, 4, 6, 8 Graph:(A straight line passing through the origin (0, 0) with a slope of 2) on-Function Example: Table: Representing a non-function (relation) using a table: x | y 1 | 2 1 | 4 2 | 3 3 } 2 Map: Representing a non-function (relation) using a mapping: Input (x): 1, 1, 2, 3 Output (y): 2, 4, 3, 2 Graph: (Has multiple y-values for each x-value, indicating that the relation does not satisfy the function property )(each input corresponds to exactly one output).

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Answer 1

It looks like you're providing examples that illustrate the difference between a function and a non-function (relation). Below, I summarize the examples you've given, along with explanations:

Function Example: \( f(x) = 2x \)

Table Representation:

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

Mapping Representation:

• Input \( x \): 1, 2, 3, 4
• Output \( f(x) \): 2, 4, 6, 8

Graph Representation:

• The graph is a straight line that passes through the origin (0, 0) with a slope of 2. It indicates a linear relationship where each input \( x \) corresponds to exactly one output \( f(x) \).

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Non-Function Example (Relation)

Table Representation:

| \( x \) | \( y \) | |---------|---------| | 1 | 2 | | 1 | 4 | | 2 | 3 | | 3 | 2 |

Mapping Representation:

• Input \( x \): 1, 1, 2, 3
• Output \( y \): 2, 4, 3, 2

Graph Representation:

• The graph indicates multiple \( y \)-values associated with the same \( x \)-value (e.g., \( x = 1 \) maps to both 2 and 4). This violates the definition of a function, where each input must correspond to exactly one output.

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Summary

• A function has a unique output for each input, which can be represented clearly in tables, mappings, and graphs.
• A non-function (relation) can have multiple outputs for the same input, as illustrated in the tables, mappings, and graphs provided.

If you have any specific questions or need further assistance with these topics, feel free to ask!

Answer 2

The first example is the graph. It is not a function because it doesn't pass the vertical line test. The second example is the table of order pairs. This is a function because every input has its own unique output. And it does not repeat any input values. This is a one-to-one function because it doesn't repeat any outputs. The third example is the mapping diagram. This example is not a function because the input "12" has two outputs. If it did not have two values, then it would be a function.
11/3/2024 5:26:55 PM by Nevaeh Chavez-Blanton
The mapping diagram is not a function, because the input value of 12 points to two outputs: 8 and 2. The table is a function, because it has a 1 to 1 relationship. The graph is not a function, because the value of 4 has multiple values of y.
11/4/2024 6:00:34 PM by Trulen Hatley
The Table, is a function due to it not having any repeating numbers within the table. This is the same for the mapping diagram, because it has no repeating numbers. The graph is not a function due to it failing the vertical line test .
11/6/2024 10:04:57 AM by Grace Housler
The graph is a non-function because it shares x values and does not pass the vertical line check. The table is a function because it has no repeating inputs/outputs. The mapping diagram is a non-function because one of the inputs has two outputs.
11/14/2024 3:06:57 PM by Jaylee Lanari
The mapping diagram is a function due to non of the inputs are the same, the table is also a function for the same reason, however the graph is not a function due to it not passing the vertical line test.
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Answer 3

It appears there is some confusion in assigning function properties to the different representations (graph, table, and mapping diagram). Let's clarify these points based on the provided comments:

1. Graph Representation

• Vertical Line Test: A graph represents a function if any vertical line drawn through the graph intersects it at most once. If the graph fails this test, it is not considered a function.
• Your Comments: You correctly identified that if the graph fails the vertical line test, it is a non-function. For instance, if there are multiple \( y \)-values corresponding to a single \( x \)-value, the graph does not reflect a function.

2. Table Representation

• Unique Inputs: A table represents a function if every input (or \( x \)-value) has a unique output (or \( y \)-value). If an input repeats with different outputs, it is not a function.
• Your Comments: You correctly pointed out that if there are no repeating \( x \)-values and each one has a unique \( y \)-value, this table would represent a function. If it also shows a one-to-one relationship (no two inputs produce the same output), it is indeed a one-to-one function.

3. Mapping Diagram

• Unique Mappings: A mapping diagram illustrates the connection between domains (inputs) and ranges (outputs). If one input points to more than one output, it does not satisfy the definition of a function.
• Your Comments: If the mapping diagram shows that a single input (like 12) has multiple outputs (like 8 and 2), then it is indeed a non-function. Conversely, if each input clearly points to a single unique output, it functions correctly.

Summary

• Graph: Determine if it passes the vertical line test to classify it as a function or non-function.
• Table: Must have unique inputs for it to be a function.
• Mapping Diagram: Should not have any inputs that point to multiple outputs to be a function.

Final Thoughts

It seems several users provided insights aligning with these principles, but there might be some contradictions regarding the classifications of the graph, table, and mapping. Each representation must be evaluated carefully based on the established definitions to avoid confusion. If there are specific examples or points you'd like to discuss further, let me know!