Select The Correct Answer.Which Relation Is Also A Function?A.${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 4 & -7 \ \hline 5 & -3 \ \hline 6 & 3 \ \hline 7 & 6 \ \hline \end{tabular} } B . B. B . [ \begin{tabular}{|c|c|} \hline X X X

Mar 13, 2025 by ADMIN 246 views

**Determining a Function: A Mathematical Analysis**

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 is a special type of relation where each input is associated with exactly one output. In this article, we will explore two relations and determine which one is also a function.

Relation A

The first relation is given by the following table:

$x$	$y$
4	-7
5	-3
6	3
7	6

To determine if this relation is a function, we need to check if each input is associated with exactly one output. Looking at the table, we can see that each value of $x$ is associated with a unique value of $y$ . For example, when $x=4$ , the corresponding value of $y$ is -7, and when $x=5$ , the corresponding value of $y$ is -3. This suggests that Relation A is a function.

Relation B

The second relation is given by the following table:

$x$	$y$
4	-7
5	-3
6	3
7	6
8	-7

At first glance, Relation B appears to be similar to Relation A. However, upon closer inspection, we notice that the value of $y$ for $x=8$ is the same as the value of $y$ for $x=4$ . This means that the input $x=8$ is associated with the output $y=-7$ , which is the same output associated with the input $x=4$ . This violates the definition of a function, which requires each input to be associated with exactly one output.

Conclusion

Based on our analysis, we can conclude that Relation A is a function, while Relation B is not. Relation A satisfies the definition of a function, as each input is associated with exactly one output. In contrast, Relation B does not satisfy the definition of a function, as the input $x=8$ is associated with the same output $y=-7$ as the input $x=4$ .

Why is this important?

Understanding functions is crucial in mathematics and other fields, such as science and engineering. Functions are used to model real-world phenomena, such as the relationship between the speed of an object and its kinetic energy. In computer science, functions are used to write efficient and modular code. In economics, functions are used to model the behavior of markets and economies.

Types of Functions

There are several types of functions, including:

Linear functions: These are functions that can be written in the form $y=mx+b$ , where $m$ and $b$ are constants.
Quadratic functions: These are functions that can be written in the form $y=ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants.
Polynomial functions: These are functions that can be written in the form $y=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ , where $a_n$ , $a_{n-1}$ , ..., $a_1$ , and $a_0$ are constants.
Rational functions: These are functions that can be written in the form $y=\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials.

Real-World Applications

Functions have numerous real-world applications, including:

Physics: Functions are used to model the motion of objects, including the relationship between speed and kinetic energy.
Engineering: Functions are used to design and optimize systems, such as bridges and buildings.
Economics: Functions are used to model the behavior of markets and economies.
Computer Science: Functions are used to write efficient and modular code.

Conclusion

Introduction

Functions are a fundamental concept in mathematics and other fields. In our previous article, we explored the definition of a function and analyzed two relations to determine which one is also a function. In this article, we will answer some frequently asked questions about functions to help you better understand this important concept.

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 is a special type of relation where each input is associated with exactly one output.

Q: What are the key characteristics of a function?

A: The key characteristics of a function are:

Each input is associated with exactly one output: This means that for each value of the input, there is only one corresponding value of the output.
The output depends only on the input: This means that the output is determined solely by the input, and not by any other factor.
The function is a relation: This means that the function is a set of ordered pairs, where each pair consists of an input and an output.

Q: What are some examples of functions?

A: Some examples of functions include:

Linear functions: These are functions that can be written in the form $y=mx+b$ , where $m$ and $b$ are constants.
Quadratic functions: These are functions that can be written in the form $y=ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants.
Polynomial functions: These are functions that can be written in the form $y=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ , where $a_n$ , $a_{n-1}$ , ..., $a_1$ , and $a_0$ are constants.
Rational functions: These are functions that can be written in the form $y=\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials.

Q: What are some examples of non-functions?

A: Some examples of non-functions include:

A relation where each input is associated with more than one output: This is known as a many-to-one relation.
A relation where the output depends on more than one input: This is known as a many-to-many relation.
A relation where the output is not determined solely by the input: This is known as a non-deterministic relation.

Q: How do functions relate to real-world phenomena?

A: Functions are used to model real-world phenomena, such as the relationship between the speed of an object and its kinetic energy. In physics, functions are used to describe the motion of objects, including the relationship between position, velocity, and acceleration. In economics, functions are used to model the behavior of markets and economies.

Q: What are some common applications of functions?

A: Some common applications of functions include:

Physics: Functions are used to model the motion of objects, including the relationship between speed and kinetic energy.
Engineering: Functions are used to design and optimize systems, such as bridges and buildings.
Economics: Functions are used to model the behavior of markets and economies.
Computer Science: Functions are used to write efficient and modular code.

Q: How can I determine if a relation is a function?

A: To determine if a relation is a function, you can use the following steps:

Check if each input is associated with exactly one output: If each input is associated with more than one output, then the relation is not a function.
Check if the output depends only on the input: If the output depends on more than one input, then the relation is not a function.
Check if the relation is a relation: If the relation is not a set of ordered pairs, then it is not a function.

Conclusion

In conclusion, functions are an essential concept in mathematics and other fields. Understanding functions is crucial for modeling real-world phenomena and writing efficient code. In this article, we answered some frequently asked questions about functions to help you better understand this important concept. We hope this guide has been helpful in your understanding of functions!