Function Domain And Range Calculation For Y=-2/3x+7

In the realm of mathematics, understanding functions is paramount. Functions serve as the bedrock for countless mathematical concepts, from basic algebra to advanced calculus. One of the fundamental aspects of a function is its domain, which represents the set of all possible input values. Conversely, the range of a function encompasses the set of all possible output values generated by the function for the given domain. This comprehensive guide delves into the concept of function domains and ranges, providing a step-by-step approach to solving problems involving them. We will use a specific example to illustrate the process: given the function y = -2/3 x + 7 and the domain {-12, -6, 3, 15}, we will complete a table to map the x values (domain) to their corresponding y values (range).

Before we dive into the problem-solving process, let's solidify our understanding of function domains and ranges. The domain of a function is the set of all possible input values (often represented by the variable x) that will produce a valid output. In simpler terms, it's the set of numbers you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number. The range, on the other hand, is the set of all possible output values (often represented by the variable y) that the function can produce when you input the values from the domain. It's the set of all the results you get after applying the function to the domain. Understanding the domain and range is crucial for comprehending the behavior of a function and its limitations. For instance, knowing the domain helps us identify which input values are permissible, while the range tells us the possible output values we can expect. This knowledge is essential in various applications, such as graphing functions, solving equations, and modeling real-world phenomena.

Now, let's tackle the problem at hand. We are given the function y = -2/3 x + 7 and the domain {-12, -6, 3, 15}. Our goal is to complete a table that maps each x value in the domain to its corresponding y value. This involves substituting each x value into the function and calculating the resulting y value. Let's break down the process step-by-step:

1. Start with the first x value: In our case, the first x value in the domain is -12. We will substitute this value into the function y = -2/3 x + 7. This gives us y = -2/3 (-12) + 7.
2. Simplify the expression: Now, we need to simplify the expression to find the corresponding y value. First, we multiply -2/3 by -12. Remember that multiplying two negative numbers results in a positive number. So, -2/3 * -12 = 8. Now our equation looks like this: y = 8 + 7.
3. Calculate the y value: Finally, we add 8 and 7 to get y = 15. This means that when x is -12, y is 15. We can now fill in the first row of our table: when x = -12, y = 15.
4. Repeat for the remaining x values: We repeat the same process for the remaining x values in the domain: -6, 3, and 15. This involves substituting each value into the function y = -2/3 x + 7 and simplifying to find the corresponding y value. For x = -6, we have y = -2/3 (-6) + 7 = 4 + 7 = 11. For x = 3, we have y = -2/3 (3) + 7 = -2 + 7 = 5. And for x = 15, we have y = -2/3 (15) + 7 = -10 + 7 = -3.
5. Complete the table: After calculating the y values for all x values in the domain, we can complete the table. The completed table will show the mapping between each x value in the domain and its corresponding y value in the range. This table provides a visual representation of the function's behavior for the given domain.

Based on the calculations above, the completed table is as follows:

This table clearly shows the relationship between the x values in the domain and the corresponding y values in the range for the function y = -2/3 x + 7.

This exercise highlights the importance of understanding function domains and ranges. By knowing the domain, we can determine the valid input values for a function, and by knowing the range, we can understand the possible output values. This knowledge is essential for various mathematical tasks, including:

• Graphing functions: The domain and range help us determine the extent of the graph along the x-axis and y-axis, respectively. They tell us where the graph starts and ends, and what values it can take.
• Solving equations: Understanding the range of a function can help us determine whether an equation involving that function has a solution. If the desired output value is not within the range, then there is no solution.
• Modeling real-world phenomena: Functions are used to model various real-world situations, such as the trajectory of a projectile or the growth of a population. Understanding the domain and range in these contexts helps us interpret the model and make meaningful predictions. For example, the domain might represent the time interval over which the model is valid, and the range might represent the possible values of the quantity being modeled.

To further solidify your understanding, try solving these practice problems:

1. Given the function y = x^2 - 4 and the domain {-2, -1, 0, 1, 2}, complete a table showing the corresponding y values.
2. Given the function y = √(x + 5) and the domain {-5, -4, -1, 4, 11}, complete a table showing the corresponding y values.
3. Determine the range of the function y = 3x - 2 for the domain {-1, 0, 1, 2, 3}.

Mastering the concepts of function domains and ranges is crucial for success in mathematics. By understanding how to determine the domain and range of a function, you gain valuable insights into its behavior and limitations. This knowledge empowers you to solve a wide range of mathematical problems and apply functions effectively in various contexts. Remember to practice regularly and break down complex problems into smaller, manageable steps. With dedication and effort, you can conquer any mathematical challenge that comes your way. In summary, the domain is the set of all possible x values, and the range is the set of all possible y values. By substituting x values from the domain into the function, we can calculate the corresponding y values and determine the range. This process is fundamental to understanding and working with functions in mathematics and beyond. Keep practicing, and you'll become a function domain and range expert in no time!