The Ordered Pairs Below Represent A Relation:${ (-1, 4), (0, 6), (0, 8), (2, 10) }$Part A: Create A Mapping Diagram To Represent The Relation.Part B: Explain Why The Relation Is A Function Or Is Not A Function.
14. The Ordered Pairs Below Represent a Relation: A Mapping Diagram and Function Analysis
Part A: Creating a Mapping Diagram to Represent the Relation
A relation is a set of ordered pairs that describe the relationship between two variables. In this problem, we are given the ordered pairs (-1, 4), (0, 6), (0, 8), and (2, 10). To create a mapping diagram, we need to represent each ordered pair as a point on a coordinate plane. The x-coordinate of each point represents the first element of the ordered pair, and the y-coordinate represents the second element.
Mapping Diagram:
| x | y |
|---|---|
| -1 | 4 |
| 0 | 6 |
| 0 | 8 |
| 2 | 10 |
Visual Representation:
Imagine a coordinate plane with the x-axis on the bottom and the y-axis on the left. We can plot each ordered pair as a point on the plane:
- (-1, 4) is plotted 1 unit to the left of the y-axis and 4 units up from the x-axis.
- (0, 6) is plotted on the y-axis and 6 units up from the x-axis.
- (0, 8) is plotted on the y-axis and 8 units up from the x-axis.
- (2, 10) is plotted 2 units to the right of the y-axis and 10 units up from the x-axis.
Part B: Explaining Why the Relation is a Function or is Not a Function
A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). In other words, a function is a relation where each input corresponds to exactly one output. To determine if the given relation is a function, we need to check if each x-coordinate corresponds to exactly one y-coordinate.
Analyzing the Relation:
Looking at the ordered pairs, we can see that the x-coordinate -1 corresponds to the y-coordinate 4. This means that when x = -1, y = 4.
However, when x = 0, we have two ordered pairs: (0, 6) and (0, 8). This means that when x = 0, y can be either 6 or 8. Since there are multiple y-coordinates for the same x-coordinate, the relation is not a function.
Conclusion:
Based on the analysis, we can conclude that the given relation is not a function. This is because each x-coordinate does not correspond to exactly one y-coordinate. Instead, some x-coordinates correspond to multiple y-coordinates, which is a characteristic of a relation that is not a function.
Key Takeaways:
- A relation is a set of ordered pairs that describe the relationship between two variables.
- A function is a relation where each input corresponds to exactly one output.
- To determine if a relation is a function, we need to check if each x-coordinate corresponds to exactly one y-coordinate.
- If multiple y-coordinates correspond to the same x-coordinate, the relation is not a function.
Real-World Applications:
Understanding functions and relations is crucial in many real-world applications, such as:
- Modeling population growth and decline
- Analyzing financial data and predicting stock prices
- Designing electronic circuits and computer systems
- Solving optimization problems in fields like engineering and economics
By recognizing the characteristics of functions and relations, we can better understand and analyze complex systems, making informed decisions and predictions in various fields.
14. The Ordered Pairs Below Represent a Relation: A Mapping Diagram and Function Analysis - Q&A
Q: What is a relation in mathematics?
A: A relation is a set of ordered pairs that describe the relationship between two variables. It is a way to represent the connection between two sets of values.
Q: What is a function in mathematics?
A: A function is a relation where each input corresponds to exactly one output. In other words, a function is a relation where each x-coordinate corresponds to exactly one y-coordinate.
Q: How do I determine if a relation is a function or not?
A: To determine if a relation is a function, you need to check if each x-coordinate corresponds to exactly one y-coordinate. If multiple y-coordinates correspond to the same x-coordinate, the relation is not a function.
Q: What is a mapping diagram?
A: A mapping diagram is a visual representation of a relation, where each ordered pair is plotted as a point on a coordinate plane. The x-coordinate of each point represents the first element of the ordered pair, and the y-coordinate represents the second element.
Q: How do I create a mapping diagram?
A: To create a mapping diagram, you need to plot each ordered pair as a point on a coordinate plane. The x-axis represents the first element of the ordered pair, and the y-axis represents the second element.
Q: What is the difference between a relation and a function?
A: A relation is a set of ordered pairs that describe the relationship between two variables, while a function is a relation where each input corresponds to exactly one output.
Q: Can a relation have multiple y-coordinates for the same x-coordinate?
A: Yes, a relation can have multiple y-coordinates for the same x-coordinate. However, if a relation has multiple y-coordinates for the same x-coordinate, it is not a function.
Q: Can a function have multiple x-coordinates for the same y-coordinate?
A: No, a function cannot have multiple x-coordinates for the same y-coordinate. By definition, a function is a relation where each input corresponds to exactly one output.
Q: What are some real-world applications of functions and relations?
A: Functions and relations are used in many real-world applications, such as:
- Modeling population growth and decline
- Analyzing financial data and predicting stock prices
- Designing electronic circuits and computer systems
- Solving optimization problems in fields like engineering and economics
Q: Why is it important to understand functions and relations?
A: Understanding functions and relations is crucial in many fields, as it allows us to analyze and model complex systems, make informed decisions, and predict outcomes.
Q: Can you provide an example of a relation that is not a function?
A: Yes, consider the relation {(1, 2), (1, 3), (2, 4)}. This relation is not a function because the x-coordinate 1 corresponds to two different y-coordinates, 2 and 3.
Q: Can you provide an example of a function?
A: Yes, consider the relation {(1, 2), (2, 4), (3, 6)}. This relation is a function because each x-coordinate corresponds to exactly one y-coordinate.
Q: How do I determine if a relation is a one-to-one function?
A: To determine if a relation is a one-to-one function, you need to check if each x-coordinate corresponds to exactly one y-coordinate, and if each y-coordinate corresponds to exactly one x-coordinate.
Q: What is a one-to-one function?
A: A one-to-one function is a function where each x-coordinate corresponds to exactly one y-coordinate, and each y-coordinate corresponds to exactly one x-coordinate.