Math Problem Statement

Are they functions? Analyze the given relations, table of values, graph, and mapping diagram to determine if they represent functions.

Solution

Let’s analyze each case to determine whether they are functions:

5. Relation: {(3, 7), (4, 7), (5, 7), (6, 7)}

  • Analysis: In this set of ordered pairs, each input xx has exactly one output yy. Specifically:
    • 373 \to 7
    • 474 \to 7
    • 575 \to 7
    • 676 \to 7
  • Conclusion: This is a function because no xx-value is repeated with different yy-values.

6. Table of values:

xx-7-5-75
yy13-113
  • Analysis: In this table:
    • The x=7x = -7 is paired with both y=1y = 1 and y=1y = -1. This means that the same input (x=7x = -7) gives two different outputs.
  • Conclusion: This is not a function because an input cannot have more than one output.

7. Graph:

  • Analysis: This is a vertical set of arrows, with one arrow per xx-coordinate pointing to various yy-values. Using the vertical line test:
    • A vertical line passes through more than one point on the graph (e.g., the line at x=0x = 0 crosses arrows pointing to different yy-values).
  • Conclusion: This is not a function because the vertical line test fails.

8. Graph:

  • Analysis: This graph shows a horizontal line at y=2y = 2. Using the vertical line test:
    • A vertical line intersects the graph at only one point for any xx-value.
  • Conclusion: This is a function because the vertical line test is satisfied.

9. Mapping diagram:

  • Analysis: In the diagram:
    • x=5x = 5 maps to both y=7y = 7 and y=12y = 12.
  • Conclusion: This is not a function because the same input (x=5x = 5) has multiple outputs.

Summary of Results:

  1. 5: Function
  2. 6: Not a function
  3. 7: Not a function
  4. 8: Function
  5. 9: Not a function

Do you want a more detailed explanation or have questions?
Here are 5 related questions to deepen your understanding:

  1. What is the formal definition of a function?
  2. How can you apply the vertical line test to any graph?
  3. Why does having one xx-value map to multiple yy-values disqualify a relation as a function?
  4. How would you represent a relation that is not a function in a graph or table?
  5. Can all mappings be visualized effectively using diagrams or graphs?

Tip: When analyzing whether a relation is a function, always check if each input (xx) maps to only one output (yy).

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Math Problem Analysis

Mathematical Concepts

Functions
Relations
Vertical Line Test
Mappings

Formulas

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Theorems

Vertical Line Test

Suitable Grade Level

Grades 8-10