Determine whether the given relation is a function or not

Oct 13, 2025 by windailys.com 24 views

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 determine whether the given relation is a function or not.

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. A function is often represented as a mapping from the domain to the range, where each input in the domain is associated with exactly one output in the range.

Properties of a Function

A function must satisfy the following properties:

Domain: A function has a domain, which is the set of all possible input values.
Range: A function has a range, which is the set of all possible output values.
One-to-One: A function is one-to-one if each input is associated with exactly one output.
Onto: A function is onto if every output value in the range is associated with at least one input value in the domain.

The Given Relation

The given relation is defined as:

${ Y = \left\{ \begin{array}{ll} x, & x \leq 3 \\ -x, & x \geq 3 \end{array} \right. }$

This relation is defined in two parts: one for inputs less than or equal to 3, and one for inputs greater than or equal to 3.

Is the Given Relation a Function?

To determine whether the given relation is a function or not, we need to check if it satisfies the properties of a function.

Domain: The domain of the given relation is all real numbers, since there are no restrictions on the input values.
Range: The range of the given relation is also all real numbers, since the output values can be any real number.
One-to-One: The given relation is not one-to-one, since each input value is associated with two output values: one for inputs less than or equal to 3, and one for inputs greater than or equal to 3.
Onto: The given relation is not onto, since not every output value in the range is associated with at least one input value in the domain.

Conclusion

Based on the properties of a function, we can conclude that the given relation is not a function. It does not satisfy the properties of a function, since it is not one-to-one and not onto.

Why is the Given Relation Not a Function?

The given relation is not a function because it does not satisfy the properties of a function. Specifically, it is not one-to-one, since each input value is associated with two output values. This means that the given relation is not a well-defined function, and it does not have a clear and consistent mapping from the domain to the range.

Real-World Examples

In real-world applications, functions are used to model relationships between variables. For example, a function can be used to model the relationship between the amount of money spent on a product and the price of the product. In this case, the function would take the amount of money spent as input and return the price of the product as output.

Conclusion

In conclusion, the given relation is not a function because it does not satisfy the properties of a function. It is not one-to-one and not onto, which means that it does not have a clear and consistent mapping from the domain to the range. This is an important concept in mathematics, and it has many real-world applications.

References

[1] "Functions" by Khan Academy
[2] "Relations and Functions" by Math Open Reference
[3] "Functions" by Wolfram MathWorld

Glossary

Domain: The set of all possible input values.
Range: The set of all possible output values.
One-to-One: A function is one-to-one if each input is associated with exactly one output.
Onto: A function is onto if every output value in the range is associated with at least one input value in the domain.
Determine whether the given relation is a function or not ===========================================================

Q&A

Q: What is a function in mathematics?

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.

Q: What are the properties of a function?

A: A function must satisfy the following properties:

Domain: A function has a domain, which is the set of all possible input values.
Range: A function has a range, which is the set of all possible output values.
One-to-One: A function is one-to-one if each input is associated with exactly one output.
Onto: A function is onto if every output value in the range is associated with at least one input value in the domain.

Q: Is the given relation a function?

A: No, the given relation is not a function. It does not satisfy the properties of a function, since it is not one-to-one and not onto.

Q: Why is the given relation not a function?

A: The given relation is not a function because it does not satisfy the properties of a function. Specifically, it is not one-to-one, since each input value is associated with two output values. This means that the given relation is not a well-defined function, and it does not have a clear and consistent mapping from the domain to the range.

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

A: Functions are used to model relationships between variables in many real-world applications. For example, a function can be used to model the relationship between the amount of money spent on a product and the price of the product. In this case, the function would take the amount of money spent as input and return the price of the product as output.

Q: How can I determine if a relation is a function or not?

A: To determine if a relation is a function or not, you need to check if it satisfies the properties of a function. Specifically, you need to check if the relation is one-to-one and onto.

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

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

Not checking if the relation is one-to-one and onto.
Not defining the domain and range of the function.
Not using a consistent mapping from the domain to the range.

Q: How can I practice working with functions?

A: You can practice working with functions by:

Solving problems that involve functions.
Working with different types of functions, such as linear and quadratic functions.
Using online resources, such as math websites and video tutorials.

Q: What are some advanced topics in functions?

A: Some advanced topics in functions include:

Inverse functions: These are functions that "undo" each other.
Composite functions: These are functions that are composed of other functions.
Parametric functions: These are functions that are defined in terms of parameters.

Q: How can I apply functions to real-world problems?

A: You can apply functions to real-world problems by:

Modeling relationships between variables using functions.
Using functions to make predictions and forecasts.
Using functions to optimize and minimize costs.

Conclusion

In conclusion, functions are an important concept in mathematics, and they have many real-world applications. By understanding the properties of functions and how to work with them, you can apply functions to solve problems and make predictions in many different fields.