Understanding the Domain and Range of a Linear Function

Oct 13, 2025 by miodeya.com 17 views

Understanding the Domain and Range of a Linear Function

In mathematics, a linear function is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In this article, we will explore how to find the domain and range of a linear function.

Domain of a Linear Function

The domain of a linear function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is defined. To find the domain of a linear function, we need to consider the values of x that make the function undefined.

Finding the Domain of a Linear Function

To find the domain of a linear function, we need to consider the values of x that make the function undefined. A linear function is undefined when the denominator of the fraction is equal to zero. In the case of the function y = -\frac{2}{5} x + 7, the denominator is 5, which is never equal to zero. Therefore, the domain of the function is all real numbers.

Range of a Linear Function

The range of a linear function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible values of y for which the function is defined. To find the range of a linear function, we need to consider the values of y that the function can take.

Finding the Range of a Linear Function

To find the range of a linear function, we need to consider the values of y that the function can take. A linear function can take any value between the minimum and maximum values of the function. In the case of the function y = -\frac{2}{5} x + 7, the minimum value of the function is -\infty and the maximum value is \infty. Therefore, the range of the function is all real numbers.

Completing the Table

Now that we have found the domain and range of the function, we can complete the table based on the given domain.

x	y
-12	?
-6	?
3	?
15	?

To complete the table, we need to plug in the values of x into the function y = -\frac{2}{5} x + 7.

Calculating y for x = -12

y = -\frac{2}{5} (-12) + 7 y = \frac{24}{5} + 7 y = 4.8 + 7 y = 11.8

Calculating y for x = -6

y = -\frac{2}{5} (-6) + 7 y = \frac{12}{5} + 7 y = 2.4 + 7 y = 9.4

Calculating y for x = 3

y = -\frac{2}{5} (3) + 7 y = -\frac{6}{5} + 7 y = -1.2 + 7 y = 5.8

Calculating y for x = 15

y = -\frac{2}{5} (15) + 7 y = -6 + 7 y = 1

Completed Table

x	y
-12	11.8
-6	9.4
3	5.8
15	1

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the domain and range of a linear function.

Q: What is the domain of a linear function?

A: The domain of a linear function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is defined.

Q: How do I find the domain of a linear function?

A: To find the domain of a linear function, you need to consider the values of x that make the function undefined. A linear function is undefined when the denominator of the fraction is equal to zero. In the case of the function y = -\frac{2}{5} x + 7, the denominator is 5, which is never equal to zero. Therefore, the domain of the function is all real numbers.

Q: What is the range of a linear function?

A: The range of a linear function is the set of all possible output values for which the function is defined. In other words, it is the set of all possible values of y for which the function is defined.

Q: How do I find the range of a linear function?

A: To find the range of a linear function, you need to consider the values of y that the function can take. A linear function can take any value between the minimum and maximum values of the function. In the case of the function y = -\frac{2}{5} x + 7, the minimum value of the function is -\infty and the maximum value is \infty. Therefore, the range of the function is all real numbers.

Q: How do I complete the table for a linear function?

A: To complete the table for a linear function, you need to plug in the values of x into the function. For example, if the function is y = -\frac{2}{5} x + 7 and the domain is {-12, -6, 3, 15}, you need to calculate the values of y for each value of x.

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

A: The domain of a linear function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values for which the function is defined.

Q: Can a linear function have a domain of all real numbers?

A: Yes, a linear function can have a domain of all real numbers. For example, the function y = -\frac{2}{5} x + 7 has a domain of all real numbers.

Q: Can a linear function have a range of all real numbers?

A: Yes, a linear function can have a range of all real numbers. For example, the function y = -\frac{2}{5} x + 7 has a range of all real numbers.

Q: How do I graph a linear function?

A: To graph a linear function, you need to plot the points on the coordinate plane and draw a line through them. For example, if the function is y = -\frac{2}{5} x + 7, you need to plot the points (-12, 11.8), (-6, 9.4), (3, 5.8), and (15, 1) on the coordinate plane and draw a line through them.

Conclusion

In this article, we have answered some frequently asked questions about the domain and range of a linear function. We have also provided examples and explanations to help you understand the concepts.