Select The Correct Answer.Which Of The Following Sets Of Ordered Pairs Represents A Function?A. { ( − 8 , − 14 ) , ( − 7 , − 12 ) , ( − 6 , − 10 ) , ( − 5 , − 8 ) } \{(-8,-14),(-7,-12),(-6,-10),(-5,-8)\} {( − 8 , − 14 ) , ( − 7 , − 12 ) , ( − 6 , − 10 ) , ( − 5 , − 8 )} B. { ( − 4 , − 14 ) , ( − 9 , − 12 ) , ( − 6 , − 10 ) , ( − 9 , − 8 ) } \{(-4,-14),(-9,-12),(-6,-10),(-9,-8)\} {( − 4 , − 14 ) , ( − 9 , − 12 ) , ( − 6 , − 10 ) , ( − 9 , − 8 )} C. { ( 8 , − 2 ) , ( 9 , − 1 ) , ( 10 , 2 ) , ( 8 , − 10 ) } \{(8,-2),(9,-1),(10,2),(8,-10)\} {( 8 , − 2 ) , ( 9 , − 1 ) , ( 10 , 2 ) , ( 8 , − 10 )} D.

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 relationship between two sets of values, where each input value corresponds to exactly one output value. In this article, we will explore the concept of functions and determine which of the given sets of ordered pairs represents a function.

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 relationship between two sets of values, where each input value corresponds to exactly one output value. In other words, for every input value, there is only one corresponding output value.

Key Characteristics of a Function

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

1. Each input value corresponds to exactly one output value: This means that for every input value, there is only one corresponding output value.
2. No two different input values have the same output value: This means that if two input values are different, their corresponding output values must also be different.

Analyzing the Given Sets of Ordered Pairs

Let's analyze the given sets of ordered pairs to determine which one represents a function.

Set A: $\{(-8,-14),(-7,-12),(-6,-10),(-5,-8)\}$

This set of ordered pairs represents a function because each input value corresponds to exactly one output value. The input values are -8, -7, -6, and -5, and the corresponding output values are -14, -12, -10, and -8, respectively. There are no two different input values with the same output value.

Set B: $\{(-4,-14),(-9,-12),(-6,-10),(-9,-8)\}$

This set of ordered pairs does not represent a function because there are two different input values (-9) with the same output value (-12 and -8). This means that the relation does not satisfy the key characteristic of a function.

Set C: $\{(8,-2),(9,-1),(10,2),(8,-10)\}$

This set of ordered pairs does not represent a function because there are two different input values (8) with the same output value (-2 and -10). This means that the relation does not satisfy the key characteristic of a function.

Set D: $\{(4,3),(5,2),(6,1),(7,0)\}$

This set of ordered pairs represents a function because each input value corresponds to exactly one output value. The input values are 4, 5, 6, and 7, and the corresponding output values are 3, 2, 1, and 0, respectively. There are no two different input values with the same output value.

Conclusion

In conclusion, the set of ordered pairs $\{(-8,-14),(-7,-12),(-6,-10),(-5,-8)\}$ represents a function because each input value corresponds to exactly one output value. The other sets of ordered pairs do not represent a function because they do not satisfy the key characteristic of a function.

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 future economic trends.
• 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 Types of Functions

There are several common types of functions, including:

• Linear functions: These are functions that have a constant rate of change, such as f(x) = 2x + 3.
• Quadratic functions: These are functions that have a parabolic shape, such as f(x) = x^2 + 2x + 1.
• Polynomial functions: These are functions that have a sum of terms, each of which is a power of x, such as f(x) = x^3 + 2x^2 + x + 1.
• Rational functions: These are functions that have a ratio of polynomials, such as f(x) = (x^2 + 1) / (x + 1).

Conclusion

In this article, we will answer some frequently asked questions about functions, including their definition, characteristics, and applications.

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 relationship between two sets of values, where each input value corresponds to exactly one output value.

Q: What are the key characteristics of a function?

A: The key characteristics of a function are:

1. Each input value corresponds to exactly one output value: This means that for every input value, there is only one corresponding output value.
2. No two different input values have the same output value: This means that if two input values are different, their corresponding output values must also be different.

Q: What are some common types of functions?

A: Some common types of functions include:

• Linear functions: These are functions that have a constant rate of change, such as f(x) = 2x + 3.
• Quadratic functions: These are functions that have a parabolic shape, such as f(x) = x^2 + 2x + 1.
• Polynomial functions: These are functions that have a sum of terms, each of which is a power of x, such as f(x) = x^3 + 2x^2 + x + 1.
• Rational functions: These are functions that have a ratio of polynomials, such as f(x) = (x^2 + 1) / (x + 1).

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 of a function. Specifically, you need to check if each input value corresponds to exactly one output value and if no two different input values have the same output value.

Q: What are some real-world applications of functions?

A: 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 future economic trends.
• 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.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on a coordinate plane that satisfy the function. You can use a graphing calculator or a computer program to help you graph the function.

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

A: A function is a relation that satisfies the key characteristics of a function, whereas a relation is a set of ordered pairs that do not necessarily satisfy these characteristics.

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

A: No, a function cannot have multiple outputs for a single input. By definition, a function is a relation where each input value corresponds to exactly one output value.

Q: Can a function be represented as a table?

A: Yes, a function can be represented as a table, where the input values are listed in one column and the corresponding output values are listed in another column.

Conclusion

In conclusion, functions are an important concept in mathematics that have numerous real-world applications. Understanding functions is essential for success in various fields, including science, engineering, economics, and computer science. By answering these frequently asked questions, we hope to have provided a better understanding of functions and their characteristics.