Including input values in the range

Oct 13, 2025 by magicalceylontours.com 14 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. The set of ordered pairs ${(-1,8), (0,3), (1,-2), (2,-7)}$ represents a function, where each input is associated with a unique output. In this article, we will explore the concept of a function and determine the range of the given set of ordered pairs.

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 is associated with a unique output. The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

The Set of Ordered Pairs

The set of ordered pairs ${(-1,8), (0,3), (1,-2), (2,-7)}$ represents a function. Each ordered pair consists of an input value and an output value. For example, the first ordered pair (-1,8) means that the input value is -1 and the output value is 8.

Understanding the Range

The range of a function is the set of all possible output values. To determine the range of the given set of ordered pairs, we need to examine the output values of each ordered pair. The output values are 8, 3, -2, and -7.

Determining the Range

To determine the range of the function, we need to identify the set of all possible output values. In this case, the output values are 8, 3, -2, and -7. Therefore, the range of the function is the set of all these output values.

The Correct Answer

Based on the analysis above, the correct answer is:

B. ${$ {y: y = -7, -2, 3, 8}$}$

This answer indicates that the range of the function is the set of all possible output values, which are -7, -2, 3, and 8.

Conclusion

In conclusion, the set of ordered pairs ${(-1,8), (0,3), (1,-2), (2,-7)}$ represents a function. The range of the function is the set of all possible output values, which are -7, -2, 3, and 8. Therefore, the correct answer is B. ${$ {y: y = -7, -2, 3, 8}$}$.

Additional Discussion

To further understand the concept of a function and its range, let's consider the following discussion points:

What is the domain of the function?

The domain of the function is the set of all possible input values. In this case, the input values are -1, 0, 1, and 2.
What is the difference between the domain and the range?

The domain is the set of all possible input values, while the range is the set of all possible output values.
How can we determine the range of a function?

To determine the range of a function, we need to examine the output values of each ordered pair and identify the set of all possible output values.

Common Mistakes

When determining the range of a function, it's common to make mistakes such as:

Including input values in the range

The range should only include output values, not input values.
Omitting output values in the range

The range should include all possible output values, not just some of them.

Real-World Applications

Understanding the concept of a function and its range has many real-world applications, such as:

Modeling real-world phenomena

Functions can be used to model real-world phenomena, such as population growth, temperature changes, and economic trends.
Making predictions

Functions can be used to make predictions about future events, such as predicting the stock market or weather patterns.
Optimizing systems

Functions can be used to optimize systems, such as finding the most efficient way to produce a product or minimize energy consumption.

Conclusion

Q&A: The Set of Ordered Pairs

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 is associated with a unique output.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values. In the given set of ordered pairs, the input values are -1, 0, 1, and 2.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values. In the given set of ordered pairs, the output values are 8, 3, -2, and -7.

Q: How can we determine the range of a function?

A: To determine the range of a function, we need to examine the output values of each ordered pair and identify the set of all possible output values.

Q: What is the difference between the domain and the range?

A: The domain is the set of all possible input values, while the range is the set of all possible output values.

Q: Can a function have multiple output values for the same input value?

A: No, a function cannot have multiple output values for the same input value. Each input value is associated with a unique output value.

Q: Can a function have no output values?

A: No, a function cannot have no output values. Each input value must be associated with at least one output value.

Q: Can a function have multiple input values for the same output value?

A: Yes, a function can have multiple input values for the same output value. This is known as a many-to-one relationship.

Q: What is the significance of the range of a function?

A: The range of a function is significant because it determines the possible output values of the function. It helps us understand the behavior of the function and make predictions about its output values.

Q: How can we use the range of a function in real-world applications?

A: We can use the range of a function in real-world applications such as modeling real-world phenomena, making predictions, and optimizing systems.

Q: Can we determine the range of a function from a graph?

A: Yes, we can determine the range of a function from a graph. The range of a function is the set of all possible output values, which can be determined by examining the graph of the function.

Q: What is the relationship between the domain and the range of a function?

A: The domain and the range of a function are related in that the domain determines the possible input values, while the range determines the possible output values.

Q: Can a function have an empty range?

A: No, a function cannot have an empty range. The range of a function must always contain at least one output value.

Q: Can a function have a range that contains only one output value?

A: Yes, a function can have a range that contains only one output value. This is known as a constant function.

Q: What is the significance of the range of a function in calculus?

A: The range of a function is significant in calculus because it determines the possible output values of the function. It helps us understand the behavior of the function and make predictions about its output values.

Conclusion

In conclusion, the set of ordered pairs ${(-1,8), (0,3), (1,-2), (2,-7)}$ represents a function. The range of the function is the set of all possible output values, which are -7, -2, 3, and 8. Understanding the concept of a function and its range has many real-world applications, such as modeling real-world phenomena, making predictions, and optimizing systems.