Each input value corresponds to only one output value

Oct 13, 2025 by miodeya.com 30 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. It is a way of describing a relationship between two sets of values. A function is often represented as a set of ordered pairs, where each ordered pair consists of an input value and a corresponding output value. In this article, we will explore three sets of ordered pairs and determine which one is not a function.

What is a Function?

A function is a relation between a set of inputs and a set of possible outputs. It is a way of describing a relationship between two sets of values. A function is often represented as a set of ordered pairs, where each ordered pair consists of an input value and a corresponding output value. For a relation to be a function, each input value must correspond to only one output value.

Set A: { (5,1), (4,1), (3,1), (2,1), (1,1) }

Let's examine the first set of ordered pairs: { (5,1), (4,1), (3,1), (2,1), (1,1) }. This set consists of five ordered pairs, where each ordered pair consists of an input value and a corresponding output value. The input values are 5, 4, 3, 2, and 1, and the corresponding output values are all 1.

Is Set A a Function?

Since each input value corresponds to only one output value, this set of ordered pairs is a function. In other words, for each input value, there is only one corresponding output value.

Set B: { (-5,-7), (-4,-6), (-3,-5), (-2,-4), (-1,-3) }

Now, let's examine the second set of ordered pairs: { (-5,-7), (-4,-6), (-3,-5), (-2,-4), (-1,-3) }. This set consists of five ordered pairs, where each ordered pair consists of an input value and a corresponding output value. The input values are -5, -4, -3, -2, and -1, and the corresponding output values are -7, -6, -5, -4, and -3.

Is Set B a Function?

Since each input value corresponds to only one output value, this set of ordered pairs is a function. In other words, for each input value, there is only one corresponding output value.

Set C: { (2,3), (2,4), (3,1), (4,2), (5,3) }

Now, let's examine the third set of ordered pairs: { (2,3), (2,4), (3,1), (4,2), (5,3) }. This set consists of five ordered pairs, where each ordered pair consists of an input value and a corresponding output value. The input values are 2, 3, 4, and 5, and the corresponding output values are 3, 4, 1, 2, and 3.

Is Set C a Function?

Since the input value 2 corresponds to two different output values, 3 and 4, this set of ordered pairs is not a function. In other words, for the input value 2, there are two corresponding output values, which violates the definition of a function.

Conclusion

In conclusion, the set of ordered pairs { (2,3), (2,4), (3,1), (4,2), (5,3) } is not a function because the input value 2 corresponds to two different output values, 3 and 4. The other two sets of ordered pairs, { (5,1), (4,1), (3,1), (2,1), (1,1) } and { (-5,-7), (-4,-6), (-3,-5), (-2,-4), (-1,-3) }, are functions because each input value corresponds to only one output value.

References

Khan Academy. (n.d.). Functions. Retrieved from https://www.khanacademy.org/math/algebra-functions/functions/v/functions
Math Is Fun. (n.d.). Functions. Retrieved from https://www.mathisfun.com/algebra/functions.html
Wikipedia. (n.d.). Function (mathematics). Retrieved from https://en.wikipedia.org/wiki/Function_(mathematics)
Q&A: Functions in Mathematics =====================================

Introduction

Functions are a fundamental concept in mathematics, and understanding them is crucial for success in various mathematical disciplines. In this article, we will answer some frequently asked questions about functions, providing a deeper understanding of this important mathematical 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. It is a way of describing a relationship between two sets of values. A function is often represented as a set of ordered pairs, where each ordered pair consists of an input value and a corresponding output value.

Q: What are the key characteristics of a function?

A: The key characteristics of a function are:

Each input value corresponds to only one output value: This means that for each input value, there is only one corresponding output value.
The output value is determined by the input value: This means that the output value is dependent on the input value.
The function is a relation between the domain and the range: This means that the function is a way of describing a relationship between the domain and the range.

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

A: A function is a relation where each input value corresponds to only one output value. A relation, on the other hand, is a set of ordered pairs where each input value may correspond to multiple output values.

Q: How do you determine if a relation is a function?

A: To determine if a relation is a function, you need to check if each input value corresponds to only one output value. If each input value corresponds to only one output value, then the relation is a function.

Q: What are some examples of functions?

A: Some examples of functions include:

Linear functions: These are functions that can be represented by a linear equation, such as f(x) = 2x + 3.
Quadratic functions: These are functions that can be represented by a quadratic equation, such as f(x) = x^2 + 2x + 1.
Polynomial functions: These are functions that can be represented by a polynomial equation, such as f(x) = x^3 + 2x^2 + x + 1.

Q: What are some examples of relations that are not functions?

A: Some examples of relations that are not functions include:

A relation where each input value corresponds to multiple output values: For example, the relation {(2,3), (2,4), (3,1), (4,2), (5,3)} is not a function because the input value 2 corresponds to two different output values, 3 and 4.
A relation where the output value is not determined by the input value: For example, the relation {(2,3), (3,4), (4,5), (5,6)} is not a function because the output value is not determined by the input value.

Q: Why are functions important in mathematics?

A: Functions are important in mathematics because they provide a way of describing a relationship between two sets of values. They are used to model real-world phenomena, such as population growth, temperature changes, and economic trends. Functions are also used to solve problems in various mathematical disciplines, such as algebra, calculus, and statistics.

Conclusion

In conclusion, functions are a fundamental concept in mathematics, and understanding them is crucial for success in various mathematical disciplines. By answering some frequently asked questions about functions, we have provided a deeper understanding of this important mathematical concept. We hope that this article has been helpful in clarifying the concept of functions and their importance in mathematics.