Finding The Range Of A Relation 12x + 6y = 24 With Domain {-4, 0, 5}
In mathematics, understanding the relationship between the domain and range of a function or relation is crucial. The domain represents the set of all possible input values (often the x-values), while the range represents the set of all possible output values (often the y-values) that result from those inputs. This article delves into the process of determining the range of a given relation when the domain is specified. We will use the example relation 12x + 6y = 24 with the domain {-4, 0, 5} to illustrate the steps involved in finding the correct range. This comprehensive guide aims to provide a clear understanding of the concepts and techniques required to solve such problems, ensuring you can confidently tackle similar questions in the future.
The concept of domain and range is fundamental in the study of functions and relations. A relation is simply a set of ordered pairs (x, y), while a function is a special type of relation where each input x is associated with exactly one output y. The domain of a relation or function is the set of all possible x-values, and the range is the set of all corresponding y-values. When we are given a relation and a specific domain, our task is to find the corresponding range by substituting each value from the domain into the relation and solving for y. This process involves algebraic manipulation and careful attention to detail to ensure accuracy. In the following sections, we will explore the steps involved in finding the range for the relation 12x + 6y = 24 with the domain {-4, 0, 5}, providing a clear and methodical approach that can be applied to other similar problems. By understanding these concepts and techniques, you will be well-equipped to handle various mathematical problems involving relations, functions, domains, and ranges.
Step-by-Step Solution: Finding the Range
To determine the range for the relation 12x + 6y = 24 with the domain {-4, 0, 5}, we need to substitute each value from the domain into the equation and solve for y. This will give us the corresponding y-values, which constitute the range. Let's break down the process step by step.
1. Substitute x = -4 into the equation:
Starting with the first value in the domain, x = -4, we substitute it into the equation 12x + 6y = 24:
12(-4) + 6y = 24
This simplifies to:
-48 + 6y = 24
Now, we need to isolate y. To do this, we add 48 to both sides of the equation:
6y = 24 + 48
6y = 72
Finally, we divide both sides by 6 to solve for y:
y = 72 / 6
y = 12
So, when x = -4, the corresponding y-value is 12. This means that the ordered pair (-4, 12) is part of the relation.
2. Substitute x = 0 into the equation:
Next, we substitute the second value from the domain, x = 0, into the equation 12x + 6y = 24:
12(0) + 6y = 24
This simplifies to:
0 + 6y = 24
6y = 24
Now, we divide both sides by 6 to solve for y:
y = 24 / 6
y = 4
So, when x = 0, the corresponding y-value is 4. This means that the ordered pair (0, 4) is part of the relation.
3. Substitute x = 5 into the equation:
Finally, we substitute the third value from the domain, x = 5, into the equation 12x + 6y = 24:
12(5) + 6y = 24
This simplifies to:
60 + 6y = 24
Now, we need to isolate y. To do this, we subtract 60 from both sides of the equation:
6y = 24 - 60
6y = -36
Finally, we divide both sides by 6 to solve for y:
y = -36 / 6
y = -6
So, when x = 5, the corresponding y-value is -6. This means that the ordered pair (5, -6) is part of the relation.
4. Determine the Range:
Now that we have found the y-values corresponding to each x-value in the domain, we can determine the range. The range is the set of all y-values we calculated:
{12, 4, -6}
Therefore, the range for the relation 12x + 6y = 24 with the domain {-4, 0, 5} is {12, 4, -6}. This matches option A in the given choices.
Analyzing the Options
Now that we have found the range by substituting the domain values into the equation, let's analyze the given options to confirm our answer and understand why the other options are incorrect.
- *A. 12, 4, -6}** are indeed 12, 4, and -6.
- B. {-4, 4, 14}: This option is incorrect. While 4 is a correct value in the range, -4 is from the domain, not the range, and 14 does not result from substituting any of the domain values into the equation.
- C. {-12, -4, 6}: This option is also incorrect. None of these values result from substituting the domain values into the equation and solving for y.
- D. {2, 4, 9}: This option is incorrect as well. While 4 is a correct value in the range, 2 and 9 do not result from substituting any of the domain values into the equation.
By analyzing the options, we can clearly see that option A is the only one that matches the range we calculated. This reinforces the importance of carefully substituting the domain values into the equation and solving for the corresponding y-values.
Key Concepts Revisited
To solidify our understanding, let's revisit the key concepts involved in this problem.
Domain and Range:
The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). In the context of a relation or function, the domain determines the set of values that can be used as inputs, while the range is the set of values that result from those inputs. Understanding the distinction between the domain and range is crucial for solving problems involving relations and functions.
Relation:
A relation is a set of ordered pairs (x, y). These pairs represent a connection or relationship between two variables. The equation 12x + 6y = 24 defines a relation between x and y. By understanding the equation, we can determine the relationship between the variables and find the corresponding y-values for given x-values.
Substitution and Solving:
The process of finding the range involves substituting the values from the domain into the equation and solving for the corresponding y-values. This requires careful algebraic manipulation and attention to detail. By following a systematic approach, we can ensure that we find the correct y-values and, consequently, the correct range.
Conclusion
In this article, we explored the process of finding the range of a relation given a specific domain. We used the example relation 12x + 6y = 24 with the domain {-4, 0, 5} to illustrate the steps involved in finding the correct range. By substituting each value from the domain into the equation and solving for y, we determined that the range is {12, 4, -6}. This process highlights the importance of understanding the concepts of domain and range, as well as the ability to perform algebraic manipulation accurately.
Furthermore, we analyzed the given options to confirm our answer and understand why the other options were incorrect. This reinforced the importance of careful calculation and attention to detail when solving mathematical problems. By revisiting the key concepts and providing a step-by-step solution, this guide aims to equip you with the knowledge and skills necessary to confidently tackle similar problems in the future. Remember, understanding the relationship between the domain and range is fundamental in mathematics, and mastering this concept will open doors to more advanced topics and applications.
