Each Answer Choice Below Represents A Relation By A Set Of Ordered Pairs. In Which Of The Answer Choices Is The Relation A Function?Select All Correct Answers.A. { {(-2,9),(3,8),(7,2),(-2,13)}$}$B.
In mathematics, a relation is a set of ordered pairs that describe the relationship between two sets of values. A function, on the other hand, is a special type of relation where each input value corresponds to exactly one output value. In this article, we will explore the concept of relations and functions, and determine which of the given answer choices represent a function.
What is a Relation?
A relation is a set of ordered pairs that describe the relationship between two sets of values. For example, the set of ordered pairs {(2, 4), (3, 6), (4, 8)} represents a relation where each input value (2, 3, 4) corresponds to an output value (4, 6, 8).
What is a Function?
A function is a special type of relation where each input value corresponds to exactly one output value. In other words, a function is a relation where each input value is mapped to a unique output value. For example, the set of ordered pairs {(2, 4), (3, 6), (4, 8)} represents a function where each input value (2, 3, 4) corresponds to a unique output value (4, 6, 8).
Determining if a Relation is a Function
To determine if a relation is a function, we need to check if each input value corresponds to exactly one output value. If an input value corresponds to more than one output value, then the relation is not a function.
Analyzing Answer Choice A
Answer choice A is given by the set of ordered pairs {(-2, 9), (3, 8), (7, 2), (-2, 13)}. To determine if this relation is a function, we need to check if each input value corresponds to exactly one output value.
- The input value -2 corresponds to two output values: 9 and 13. Since an input value cannot correspond to more than one output value, this relation is not a function.
Analyzing Answer Choice B
Answer choice B is given by the set of ordered pairs {(1, 2), (2, 3), (3, 4), (4, 5)}. To determine if this relation is a function, we need to check if each input value corresponds to exactly one output value.
- The input value 1 corresponds to the output value 2.
- The input value 2 corresponds to the output value 3.
- The input value 3 corresponds to the output value 4.
- The input value 4 corresponds to the output value 5.
Since each input value corresponds to exactly one output value, this relation is a function.
Analyzing Answer Choice C
Answer choice C is given by the set of ordered pairs {(1, 2), (2, 2), (3, 4), (4, 5)}. To determine if this relation is a function, we need to check if each input value corresponds to exactly one output value.
- The input value 1 corresponds to the output value 2.
- The input value 2 corresponds to two output values: 2 and 2. Since an input value cannot correspond to more than one output value, this relation is not a function.
Analyzing Answer Choice D
Answer choice D is given by the set of ordered pairs {(1, 2), (2, 3), (3, 4), (4, 4)}. To determine if this relation is a function, we need to check if each input value corresponds to exactly one output value.
- The input value 1 corresponds to the output value 2.
- The input value 2 corresponds to the output value 3.
- The input value 3 corresponds to the output value 4.
- The input value 4 corresponds to two output values: 4 and 4. Since an input value cannot correspond to more than one output value, this relation is not a function.
Conclusion
In conclusion, the only answer choice that represents a function is answer choice B. This is because each input value in answer choice B corresponds to exactly one output value, which is the defining characteristic of a function.
Key Takeaways
- A relation is a set of ordered pairs that describe the relationship between two sets of values.
- A function is a special type of relation where each input value corresponds to exactly one output value.
- To determine if a relation is a function, we need to check if each input value corresponds to exactly one output value.
- Answer choice B is the only answer choice that represents a function.
Final Answer
The final answer is:
- Answer choice B is the only answer choice that represents a function.
Q&A: Relations and Functions in Mathematics =====================================================
In our previous article, we explored the concept of relations and functions in mathematics. We discussed what a relation is, what a function is, and how to determine if a relation is a function. In this article, we will answer some frequently asked questions about relations and functions.
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 sets of values. A function, on the other hand, is a special type of relation where each input value corresponds to exactly one output value.
Q: How do I determine if a relation is a function?
A: To determine if a relation is a function, you need to check if each input value corresponds to exactly one output value. If an input value corresponds to more than one output value, then the relation is not a function.
Q: What is an example of a relation that is not a function?
A: An example of a relation that is not a function is the set of ordered pairs {(2, 4), (3, 6), (4, 8), (2, 10)}. In this relation, the input value 2 corresponds to two output values: 4 and 10. Since an input value cannot correspond to more than one output value, this relation is not a function.
Q: What is an example of a function?
A: An example of a function is the set of ordered pairs {(2, 4), (3, 6), (4, 8)}. In this function, each input value corresponds to exactly one output value.
Q: Can a function have multiple output values for the same input value?
A: No, a function cannot have multiple output values for the same input value. By definition, a function is a relation where each input value corresponds to exactly one output value.
Q: Can a relation have multiple input values for the same output value?
A: Yes, a relation can have multiple input values for the same output value. However, if a relation is a function, then each input value must correspond to exactly one output value.
Q: How do I represent a function in mathematical notation?
A: A function can be represented in mathematical notation using the following notation:
f(x) = y
Where f(x) is the function, x is the input value, and y is the output value.
Q: What is the domain and range of a function?
A: The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values.
Q: Can the domain and range of a function be the same?
A: Yes, the domain and range of a function can be the same. For example, the function f(x) = x has the same domain and range, which is the set of all real numbers.
Q: Can a function have a domain that is a subset of the real numbers?
A: Yes, a function can have a domain that is a subset of the real numbers. For example, the function f(x) = x^2 has a domain that is the set of all non-negative real numbers.
Q: Can a function have a range that is a subset of the real numbers?
A: Yes, a function can have a range that is a subset of the real numbers. For example, the function f(x) = x^2 has a range that is the set of all non-negative real numbers.
Conclusion
In conclusion, relations and functions are fundamental concepts in mathematics. Understanding the difference between a relation and a function, and how to determine if a relation is a function, is crucial for solving mathematical problems. We hope that this Q&A article has helped to clarify any questions you may have had about relations and functions.
Key Takeaways
- A relation is a set of ordered pairs that describe the relationship between two sets of values.
- A function is a special type of relation where each input value corresponds to exactly one output value.
- To determine if a relation is a function, you need to check if each input value corresponds to exactly one output value.
- The domain of a function is the set of all possible input values.
- The range of a function is the set of all possible output values.
Final Answer
The final answer is:
- Relations and functions are fundamental concepts in mathematics.
- Understanding the difference between a relation and a function is crucial for solving mathematical problems.