Understanding Functions and Their Domain and Range

Oct 13, 2025 by miodeya.com 103 views

Understanding Functions and Their Domain and Range

When it comes to functions, understanding the domain and range is crucial. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this article, we will explore the concept of functions and how to determine their domain and range.

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. In mathematical terms, a function is a relation between two sets, X and Y, such that each element of X is related to exactly one element of Y.

The Given Function

The given function is F = {(x, y) | x + y = 10}. This function represents a relation between two sets of values, x and y, such that their sum is equal to 10.

Determining the Domain and Range

To determine the domain and range of the function F, we need to analyze the given relation. The relation states that x + y = 10. This means that for every value of x, there is a corresponding value of y that satisfies the equation.

Domain

The domain of a function is the set of all possible input values. In this case, the domain is the set of all possible values of x that satisfy the equation x + y = 10. Since y is also a variable, we can express y in terms of x as y = 10 - x. This means that for every value of x, there is a corresponding value of y that satisfies the equation.

However, the domain is not just the set of all possible values of x. It is also the set of all possible values of x that make the function defined. In this case, the function is defined for all real numbers, since there is no restriction on the values of x.

Therefore, the domain of the function F is the set of all real numbers, denoted as R.

Range

The range of a function is the set of all possible output values. In this case, the range is the set of all possible values of y that satisfy the equation x + y = 10. Since y is expressed in terms of x as y = 10 - x, the range is also the set of all real numbers, denoted as R.

However, the range is not just the set of all possible values of y. It is also the set of all possible values of y that make the function defined. In this case, the function is defined for all real numbers, since there is no restriction on the values of y.

Therefore, the range of the function F is the set of all real numbers, denoted as R.

Conclusion

In conclusion, the domain and range of the function F = {(x, y) | x + y = 10} are both the set of all real numbers, denoted as R. This means that the function is defined for all real numbers and takes on all real numbers as output values.

Answer

The correct answer is A. Domain: All Real Numbers, Range: All Real Numbers.

Why is the other option incorrect?

Option B is incorrect because the domain and range are not just the set of all possible values of x and y, respectively. They are also the set of all possible values of x and y that make the function defined.

Option C is incorrect because the given function is indeed a function, and it does contain a domain and range. The domain is the set of all real numbers, and the range is also the set of all real numbers.

Final Thoughts

In conclusion, understanding the domain and range of a function is crucial in mathematics. The domain is the set of all possible input values, while the range is the set of all possible output values. By analyzing the given relation, we can determine the domain and range of the function F = {(x, y) | x + y = 10} and conclude that both are the set of all real numbers, denoted as R.
Frequently Asked Questions (FAQs) About Domain and Range

In the previous article, we discussed the concept of domain and range in functions and how to determine them. However, we understand that there may be more questions and doubts that need to be clarified. In this article, we will address some frequently asked questions (FAQs) about domain and range.

Q: What is the difference between domain and range?

A: The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In other words, the domain is the set of all possible x-values, and the range is the set of all possible y-values.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to analyze the given relation and identify any restrictions on the input values. For example, if the function has a denominator that cannot be zero, then the domain will exclude that value.

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

A: A function is a relation between a set of inputs (domain) and a set of possible outputs (range), where each input is related to exactly one output. A relation, on the other hand, is a set of ordered pairs that do not necessarily satisfy the function property.

Q: Can a function have a domain that is not a set of real numbers?

A: Yes, a function can have a domain that is not a set of real numbers. For example, a function with a domain of integers or a domain of complex numbers is still a function.

Q: Can a function have a range that is not a set of real numbers?

A: Yes, a function can have a range that is not a set of real numbers. For example, a function with a range of integers or a range of complex numbers is still a function.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to analyze the given relation and identify any restrictions on the output values. For example, if the function has a maximum or minimum value, then the range will be limited to that value.

Q: Can a function have a range that is empty?

A: Yes, a function can have a range that is empty. For example, a function with a domain of all real numbers but a range of only one value is still a function.

Q: Can a function have a domain and range that are the same?

A: Yes, a function can have a domain and range that are the same. For example, a function with a domain and range of all real numbers is still a function.

Q: Can a function have a domain and range that are different?

A: Yes, a function can have a domain and range that are different. For example, a function with a domain of all real numbers but a range of only positive real numbers is still a function.

Q: What is the importance of domain and range in functions?

A: The domain and range of a function are important because they help us understand the behavior of the function and its possible outputs. By analyzing the domain and range, we can determine the possible values of the function and make predictions about its behavior.

Q: How do I graph a function with a given domain and range?

A: To graph a function with a given domain and range, you need to plot the function on a coordinate plane, using the domain as the x-values and the range as the y-values. You can use a graphing calculator or software to help you graph the function.

Q: Can I have a function with a domain and range that are not intervals?

A: Yes, a function can have a domain and range that are not intervals. For example, a function with a domain of a single point or a range of a single point is still a function.

Q: Can I have a function with a domain and range that are not finite?

A: Yes, a function can have a domain and range that are not finite. For example, a function with a domain of all real numbers or a range of all real numbers is still a function.

Conclusion

In conclusion, the domain and range of a function are important concepts that help us understand the behavior of the function and its possible outputs. By analyzing the domain and range, we can determine the possible values of the function and make predictions about its behavior. We hope that this article has helped to clarify any doubts you may have had about domain and range.