Which Of The Relations Given By The Following Sets Of Ordered Pairs Is Not A Function?A. { {(5,2),(4,2),(3,2),(2,2),(1,2)}$}$B. { {(-4,-2),(-1,-1),(3,2),(3,5),(7,10)}$}$C. { {(-6,4),(-3,-1),(0,5),(1,-1),(2,3)}$}$D.

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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 relationship between two sets of values. A function is said to be a relation if each input is associated with exactly one output. In this article, we will explore which of the given sets of ordered pairs is not 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. A function is said to be a relation if each input is associated with exactly one output. In other words, for every input, there is only one corresponding output.

Definition of a Function

A function can be defined as a relation between two sets, A and B, such that for every element a in A, there is exactly one element b in B that is related to a. This can be represented mathematically as:

f: A → B

where f is the function, A is the domain, and B is the range.

Types of Relations

There are two types of relations: functions and non-functions. A function is a relation where each input is associated with exactly one output. A non-function is a relation where an input is associated with more than one output.

Example of a Function

Let's consider an example of a function. Suppose we have a set of ordered pairs:

{(2, 4), (3, 9), (4, 16), (5, 25)}

In this example, each input is associated with exactly one output. For example, the input 2 is associated with the output 4, the input 3 is associated with the output 9, and so on.

Example of a Non-Function

Now, let's consider an example of a non-function. Suppose we have a set of ordered pairs:

{(2, 4), (2, 9), (3, 16), (4, 25)}

In this example, the input 2 is associated with more than one output, namely 4 and 9. This is not a function because each input is not associated with exactly one output.

Which of the Relations Given by the Following Sets of Ordered Pairs is Not a Function?

Now, let's consider the sets of ordered pairs given in the problem statement.

A. {(5,2),(4,2),(3,2),(2,2),(1,2)}

This set of ordered pairs represents a function because each input is associated with exactly one output. The input 5 is associated with the output 2, the input 4 is associated with the output 2, and so on.

B. {(-4,-2),(-1,-1),(3,2),(3,5),(7,10)}

This set of ordered pairs does not represent a function because the input 3 is associated with more than one output, namely 2 and 5. This is not a function because each input is not associated with exactly one output.

C. {(-6,4),(-3,-1),(0,5),(1,-1),(2,3)}

This set of ordered pairs represents a function because each input is associated with exactly one output. The input -6 is associated with the output 4, the input -3 is associated with the output -1, and so on.

D.

This option is not provided in the problem statement.

Conclusion

In conclusion, the set of ordered pairs {(-4,-2),(-1,-1),(3,2),(3,5),(7,10)} is not a function because the input 3 is associated with more than one output, namely 2 and 5. This is not a function because each input is not associated with exactly one output.

References

  • [1] "Functions" by Khan Academy
  • [2] "Relations and Functions" by Math Open Reference
  • [3] "Functions and Relations" by Purplemath

Keywords

  • Function
  • Relation
  • Ordered pairs
  • Domain
  • Range
  • Non-function

Note

Introduction

In our previous article, we discussed the concept of relations and functions, and identified which of the given sets of ordered pairs is not a function. In this article, we will answer some frequently asked questions about relations and functions.

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

A: A relation is a set of ordered pairs that associates each input with one or more outputs. A function, on the other hand, is a relation where each input is associated with exactly one output.

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 each input is associated with exactly one output. If an input is associated with more than one output, then the relation 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. It is the set of all values that can be plugged into the function.

Q: What is the range of a function?

A: The range of a function is the set of all possible outputs. It is the set of all values that the function can produce.

Q: Can a function have a domain with more than one element?

A: Yes, a function can have a domain with more than one element. For example, the function f(x) = x^2 has a domain of all real numbers, which is an infinite set.

Q: Can a function have a range with more than one element?

A: Yes, a function can have a range with more than one element. For example, the function f(x) = x^2 has a range of all non-negative real numbers, which is an infinite set.

Q: Can a relation have a domain with more than one element and a range with more than one element?

A: Yes, a relation can have a domain with more than one element and a range with more than one element. For example, the relation {(2, 4), (3, 9), (4, 16)} has a domain of {2, 3, 4} and a range of {4, 9, 16}.

Q: Can a function be one-to-one?

A: Yes, a function can be one-to-one. A one-to-one function is a function where each output is associated with exactly one input.

Q: Can a function be onto?

A: Yes, a function can be onto. An onto function is a function where each output is associated with at least one input.

Q: Can a relation be one-to-one and onto?

A: Yes, a relation can be one-to-one and onto. A one-to-one and onto relation is a relation where each output is associated with exactly one input and each input is associated with exactly one output.

Conclusion

In conclusion, relations and functions are important concepts in mathematics. Understanding the difference between a relation and a function, and being able to determine if a relation is a function, is crucial in mathematics. We hope that this article has helped to clarify any questions you may have had about relations and functions.

References

  • [1] "Functions" by Khan Academy
  • [2] "Relations and Functions" by Math Open Reference
  • [3] "Functions and Relations" by Purplemath

Keywords

  • Relation
  • Function
  • Domain
  • Range
  • One-to-one
  • Onto

Note

This article is intended for educational purposes only. It is not a substitute for professional advice or guidance. If you are unsure about any mathematical concept, please consult a qualified teacher or tutor.