Select The Correct Answer.Which Relation Is A Function?A. { ( 2 , 3 ) , ( 1 , 5 ) , ( 2 , 7 ) } \{(2,3),(1,5),(2,7)\} {( 2 , 3 ) , ( 1 , 5 ) , ( 2 , 7 )} B. { ( − 1 , 5 ) , ( − 2 , 6 ) , ( − 3 , 7 ) } \{(-1,5),(-2,6),(-3,7)\} {( − 1 , 5 ) , ( − 2 , 6 ) , ( − 3 , 7 )} C. { ( 11 , 9 ) , ( 11 , 5 ) , ( 9 , 3 ) } \{(11,9),(11,5),(9,3)\} {( 11 , 9 ) , ( 11 , 5 ) , ( 9 , 3 )} D. { ( 3 , 8 ) , ( 0 , 8 ) , ( 3 , − 2 ) } \{(3,8),(0,8),(3,-2)\} {( 3 , 8 ) , ( 0 , 8 ) , ( 3 , − 2 )}

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. It is a way of describing a specific output for every input in the domain. In this article, we will explore the concept of functions and how to determine which relation is a function among the given options.

What is a Function?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a specific output for every input in the domain. In other words, for every input in the domain, there is exactly one output in the range.

Key Characteristics of a Function

To determine if a relation is a function, we need to check if it satisfies the following key characteristics:

• Each input has exactly one output: For every input in the domain, there is exactly one output in the range.
• No two different inputs have the same output: If two different inputs have the same output, then the relation is not a function.

Analyzing the Options

Let's analyze each option to determine which one is a function.

Option A: $\{(2,3),(1,5),(2,7)\}$

This relation has two inputs, 1 and 2, with different outputs, 5 and 3, respectively. However, the input 2 has two different outputs, 3 and 7. This means that the relation is not a function because it does not satisfy the key characteristic of having exactly one output for every input.

Option B: $\{(-1,5),(-2,6),(-3,7)\}$

This relation has three inputs, -1, -2, and -3, with different outputs, 5, 6, and 7, respectively. Each input has exactly one output, and no two different inputs have the same output. Therefore, this relation is a function.

Option C: $\{(11,9),(11,5),(9,3)\}$

This relation has two inputs, 11 and 9, with different outputs, 9 and 5, respectively. However, the input 11 has two different outputs, 9 and 5. This means that the relation is not a function because it does not satisfy the key characteristic of having exactly one output for every input.

Option D: $\{(3,8),(0,8),(3,-2)\}$

This relation has two inputs, 3 and 0, with different outputs, 8 and 8, respectively. However, the input 3 has two different outputs, 8 and -2. This means that the relation is not a function because it does not satisfy the key characteristic of having exactly one output for every input.

Conclusion

In conclusion, the correct answer is Option B: $\{(-1,5),(-2,6),(-3,7)\}$. This relation is a function because it satisfies the key characteristics of having exactly one output for every input and no two different inputs have the same output.

Real-World Applications of Functions

Functions have numerous real-world applications in various fields, including:

• Science: Functions are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of physical systems.
• Engineering: Functions are used to design and optimize systems, such as electronic circuits, mechanical systems, and computer algorithms.
• Economics: Functions are used to model economic systems, such as supply and demand curves, and to make predictions about economic outcomes.
• Computer Science: Functions are used to write algorithms, such as sorting and searching algorithms, and to implement data structures, such as arrays and linked lists.

Common Mistakes to Avoid

When working with functions, it's essential to avoid common mistakes, such as:

• Confusing functions with relations: A function is a specific type of relation that satisfies certain key characteristics. Make sure to distinguish between functions and relations.
• Not checking for multiple outputs: When checking if a relation is a function, make sure to check if each input has exactly one output.
• Not checking for duplicate outputs: When checking if a relation is a function, make sure to check if no two different inputs have the same output.

Tips for Working with Functions

When working with functions, here are some tips to keep in mind:

• Use clear and concise notation: Use clear and concise notation when writing functions, such as f(x) = 2x + 3.
• Check for domain and range: Make sure to check the domain and range of a function to ensure that it is well-defined.
• Use function notation: Use function notation, such as f(x), to represent functions.
• Practice, practice, practice: Practice working with functions to become proficient in using them to solve problems.

Conclusion

In conclusion, functions are an essential concept in mathematics that have numerous real-world applications. By understanding the key characteristics of functions and how to determine if a relation is a function, you can become proficient in using functions to solve problems. Remember to avoid common mistakes and follow tips for working with functions to ensure that you are using functions effectively.

Introduction

Functions are a fundamental concept in mathematics that have numerous real-world applications. In this article, we will answer some of the most frequently asked questions about functions.

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. It is a way of describing a specific output for every input in the domain.

Q: What are the key characteristics of a function?

A: The key characteristics of a function are:

• Each input has exactly one output: For every input in the domain, there is exactly one output in the range.
• No two different inputs have the same output: If two different inputs have the same output, then the relation is not a function.

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 it satisfies the key characteristics mentioned above. You can do this by:

• Checking for multiple outputs: If a relation has multiple outputs for a single input, then it is not a function.
• Checking for duplicate outputs: If a relation has duplicate outputs for different inputs, then it is not a function.

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

A: A function is a specific type of relation that satisfies certain key characteristics. A relation is a set of ordered pairs, while a function is a relation that has exactly one output for every input.

Q: Can a function have a single input with multiple outputs?

A: No, a function cannot have a single input with multiple outputs. If a function has a single input with multiple outputs, then it is not a function.

Q: Can a function have multiple inputs with the same output?

A: No, a function cannot have multiple inputs with the same output. If a function has multiple inputs with the same output, then it is not a function.

Q: What is the domain of a function?

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

Q: What is the range of a function?

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

Q: Can a function have an empty domain?

A: Yes, a function can have an empty domain. This means that the function is not defined for any input.

Q: Can a function have an empty range?

A: Yes, a function can have an empty range. This means that the function does not produce any output.

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

A: A function is a relation between a set of inputs and a set of outputs, while a mapping is a specific type of function that is used to describe a one-to-one correspondence between two sets.

Q: Can a function be represented as a table?

A: Yes, a function can be represented as a table. The table would have the input values in the first column and the corresponding output values in the second column.

Q: Can a function be represented as a graph?

A: Yes, a function can be represented as a graph. The graph would have the input values on the x-axis and the corresponding output values on the y-axis.

Conclusion

In conclusion, functions are a fundamental concept in mathematics that have numerous real-world applications. By understanding the key characteristics of functions and how to determine if a relation is a function, you can become proficient in using functions to solve problems. Remember to avoid common mistakes and follow tips for working with functions to ensure that you are using functions effectively.

Real-World Applications of Functions

Functions have numerous real-world applications in various fields, including:

• Science: Functions are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of physical systems.
• Engineering: Functions are used to design and optimize systems, such as electronic circuits, mechanical systems, and computer algorithms.
• Economics: Functions are used to model economic systems, such as supply and demand curves, and to make predictions about economic outcomes.
• Computer Science: Functions are used to write algorithms, such as sorting and searching algorithms, and to implement data structures, such as arrays and linked lists.

Common Mistakes to Avoid

When working with functions, it's essential to avoid common mistakes, such as:

• Confusing functions with relations: A function is a specific type of relation that satisfies certain key characteristics. Make sure to distinguish between functions and relations.
• Not checking for multiple outputs: When checking if a relation is a function, make sure to check if each input has exactly one output.
• Not checking for duplicate outputs: When checking if a relation is a function, make sure to check if no two different inputs have the same output.

Tips for Working with Functions

When working with functions, here are some tips to keep in mind:

• Use clear and concise notation: Use clear and concise notation when writing functions, such as f(x) = 2x + 3.
• Check for domain and range: Make sure to check the domain and range of a function to ensure that it is well-defined.
• Use function notation: Use function notation, such as f(x), to represent functions.
• Practice, practice, practice: Practice working with functions to become proficient in using them to solve problems.