Determine Whether The Following Relation Represents A Function. Give The Domain And Range For The Relation.$\{(-6,4),(-2,0),(-8,-8),(0,0)\}$Does The Given Relation Represent A Function?A. YesB. No

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function must assign to each element in the domain exactly one element in the range. In this article, we will determine whether the given relation represents a function and find the domain and range for the relation.

The Given Relation

The given relation is $\{(-6,4),(-2,0),(-8,-8),(0,0)\}$. This relation consists of four ordered pairs, where each pair represents a mapping from an input to an output.

Does the Given Relation Represent a Function?

To determine whether the given relation represents a function, we need to check if each input in the domain is assigned to exactly one output in the range. In other words, we need to check if there are any duplicate inputs with different outputs.

Upon examining the given relation, we notice that there are no duplicate inputs with different outputs. Each input in the domain is assigned to exactly one output in the range. Therefore, the given relation represents a function.

Domain and Range

The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. In this case, the domain is the set of all inputs, which is $\{-6, -2, -8, 0\}$, and the range is the set of all outputs, which is $\{4, 0, -8, 0\}$.

However, we notice that the range contains a duplicate output, which is $0$. This means that the range is not a set, but a multiset. A multiset is a set that allows duplicate elements.

To make the range a set, we can remove the duplicate element, which is $0$. The resulting range is $\{4, -8\}$.

Conclusion

In conclusion, the given relation represents a function. The domain is the set of all inputs, which is $\{-6, -2, -8, 0\}$, and the range is the set of all outputs, which is $\{4, -8\}$.

Key Takeaways

• A function is a relation between a set of inputs and a set of possible outputs.
• A function must assign to each element in the domain exactly one element in the range.
• The domain is the set of all possible inputs, while the range is the set of all possible outputs.
• A multiset is a set that allows duplicate elements.

Example Problems

1. Determine whether the relation $\{(2,3), (4,5), (2,7)\}$ represents a function.
2. Find the domain and range for the relation $\{(1,2), (3,4), (5,6)\}$.

Solutions

1. The relation $\{(2,3), (4,5), (2,7)\}$ does not represent a function because the input $2$ is assigned to two different outputs, which are $3$ and $7$.
2. The domain for the relation $\{(1,2), (3,4), (5,6)\}$ is the set of all inputs, which is $\{1, 3, 5\}$, and the range is the set of all outputs, which is $\{2, 4, 6\}$.

Further Reading

For more information on functions and relations, please refer to the following resources:

• Wikipedia: Function (mathematics)
• Khan Academy: Functions
• Math Is Fun: Functions

References

• [1] "Functions" by Khan Academy
• [2] "Functions" by Math Is Fun
• [3] "Function (mathematics)" by Wikipedia
  Q&A: Functions and Relations =============================

Introduction

In our previous article, we discussed whether the given relation represents a function and found the domain and range for the relation. In this article, we will answer some frequently asked questions about functions and relations.

Q: What is a function?

A: A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function must assign to each element in the domain exactly one element in the range.

Q: What is the difference between a function and a relation?

A: A relation is a set of ordered pairs, where each pair represents a mapping from an input to an output. A function is a relation that assigns to each element in the domain exactly one element in the range.

Q: How do I determine whether a relation represents a function?

A: To determine whether a relation represents a function, you need to check if each input in the domain is assigned to exactly one output in the range. In other words, you need to check if there are any duplicate inputs with different outputs.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible inputs.

Q: What is the range of a function?

A: The range of a function is the set of all possible outputs.

Q: Can a function have a duplicate output?

A: No, a function cannot have a duplicate output. If a function has a duplicate output, it is not a function.

Q: Can a relation have a duplicate input with different outputs?

A: Yes, a relation can have a duplicate input with different outputs. However, if a relation has a duplicate input with different outputs, it is not a function.

Q: What is a multiset?

A: A multiset is a set that allows duplicate elements.

Q: Can a function have a multiset as its range?

A: No, a function cannot have a multiset as its range. If a function has a multiset as its range, it is not a function.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, you need to examine the relation and identify the set of all possible inputs and the set of all possible outputs.

Q: What are some common mistakes to avoid when working with functions and relations?

A: Some common mistakes to avoid when working with functions and relations include:

• Assuming that a relation is a function without checking if each input is assigned to exactly one output.
• Failing to identify duplicate inputs with different outputs.
• Using a multiset as the range of a function.

Conclusion

In conclusion, functions and relations are important concepts in mathematics. By understanding the definitions and properties of functions and relations, you can avoid common mistakes and work effectively with these concepts.

Key Takeaways

• A function is a relation that assigns to each element in the domain exactly one element in the range.
• The domain of a function is the set of all possible inputs.
• The range of a function is the set of all possible outputs.
• A multiset is a set that allows duplicate elements.
• A function cannot have a duplicate output or a multiset as its range.

Example Problems

1. Determine whether the relation $\{(2,3), (4,5), (2,7)\}$ represents a function.
2. Find the domain and range for the relation $\{(1,2), (3,4), (5,6)\}$.

Solutions

1. The relation $\{(2,3), (4,5), (2,7)\}$ does not represent a function because the input $2$ is assigned to two different outputs, which are $3$ and $7$.
2. The domain for the relation $\{(1,2), (3,4), (5,6)\}$ is the set of all inputs, which is $\{1, 3, 5\}$, and the range is the set of all outputs, which is $\{2, 4, 6\}$.

Further Reading

For more information on functions and relations, please refer to the following resources:

• Wikipedia: Function (mathematics)
• Khan Academy: Functions
• Math Is Fun: Functions

References

• [1] "Functions" by Khan Academy
• [2] "Functions" by Math Is Fun
• [3] "Function (mathematics)" by Wikipedia