Domain And Range Complete The Table For Y Equals (-2/3)x Plus 7
In the realm of mathematics, understanding functions is paramount. Functions are the fundamental building blocks of many mathematical concepts, and grasping their intricacies is crucial for success in higher-level mathematics. One of the most important aspects of a function is its domain and range. The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). In this comprehensive guide, we will delve into the concept of domain and range, and how to determine the correct output values for a given domain. We will focus on a specific example: a linear function defined as y = (-2/3)x + 7, with a domain of {-12, -6, 3, 15}. By the end of this guide, you will have a solid understanding of how to calculate the range of a function for a given domain, and you will be well-equipped to tackle similar problems in the future.
Understanding Domain and Range
Before we dive into the specific example, let's first establish a clear understanding of the concepts of domain and range. The domain, in simple terms, is the set of all possible values that you can input into a function. Think of it as the input side of a vending machine – you can only insert certain types of currency (coins or bills), and the machine won't accept anything else. Similarly, a function will only accept specific values as input, and these values constitute its domain. The domain is often represented as a set of numbers, an interval, or a combination of both.
On the other hand, the range is the set of all possible output values that the function can produce. It's like the output side of the vending machine – you'll receive a specific item (a candy bar, a soda, etc.) depending on what you selected. The range is determined by the function itself and the domain. For each input value in the domain, the function performs a specific operation, and the result is the corresponding output value. The collection of all these output values forms the range of the function. Understanding the relationship between the domain and range is essential for analyzing and interpreting functions.
Key Differences Between Domain and Range
To further solidify your understanding, let's highlight the key differences between the domain and range:
- Domain: Input values (x-values), the set of all possible values that can be plugged into the function.
- Range: Output values (y-values), the set of all possible values that the function can produce.
- Domain: Independent variable, the variable whose value is chosen.
- Range: Dependent variable, the variable whose value depends on the input.
- Domain: Affects the function's definition, certain values may be excluded.
- Range: Determined by the function and the domain.
With a clear grasp of the concepts of domain and range, we can now move on to our specific example and demonstrate how to calculate the range for a given domain.
The Function and the Domain
Our specific problem involves the linear function:
y = (-2/3)x + 7
This is a linear equation, which means its graph will be a straight line. The function takes an input value (x), multiplies it by -2/3, and then adds 7 to the result. The output (y) is the value of the function at that specific input. We are given the domain of this function as the set:
{-12, -6, 3, 15}
This means we are only interested in the output values (y-values) that correspond to these specific input values (x-values). Our task is to find the range of the function for this given domain. In other words, we need to determine the y-values that result when we plug each of the x-values from the domain into the function.
Importance of Domain in Determining Range
The domain plays a crucial role in determining the range of a function. The range is not an inherent property of the function alone; it is dependent on the specified domain. If we were to change the domain, the range would likely change as well. For example, if we considered all real numbers as the domain for our function y = (-2/3)x + 7, the range would also be all real numbers. However, since we have a limited domain of only four values, our range will also be limited to a set of four corresponding values. This highlights the significance of understanding the domain when analyzing a function and its behavior.
Now, let's proceed with calculating the range for our given domain.
Calculating the Range
To find the range of the function for the given domain, we need to substitute each value from the domain into the function y = (-2/3)x + 7 and calculate the corresponding y-value. This process will give us the set of all possible output values, which is the range.
Let's take each value from the domain one by one:
-
x = -12
Substitute x = -12 into the function:
y = (-2/3)(-12) + 7
y = (24/3) + 7
y = 8 + 7
y = 15
So, when x = -12, the corresponding y-value is 15.
-
x = -6
Substitute x = -6 into the function:
y = (-2/3)(-6) + 7
y = (12/3) + 7
y = 4 + 7
y = 11
Therefore, when x = -6, the corresponding y-value is 11.
-
x = 3
Substitute x = 3 into the function:
y = (-2/3)(3) + 7
y = -2 + 7
y = 5
Thus, when x = 3, the corresponding y-value is 5.
-
x = 15
Substitute x = 15 into the function:
y = (-2/3)(15) + 7
y = -10 + 7
y = -3
Hence, when x = 15, the corresponding y-value is -3.
Now that we have calculated the y-values for each x-value in the domain, we can construct the range of the function.
The Complete Table and the Range
We can now complete the table based on our calculations:
| x | y |
|---|---|
| -12 | 15 |
| -6 | 11 |
| 3 | 5 |
| 15 | -3 |
The range of the function y = (-2/3)x + 7 for the domain {-12, -6, 3, 15} is the set of all the y-values we calculated:
{15, 11, 5, -3}
This set represents all the possible output values of the function when we input the given x-values. We have successfully determined the range by systematically substituting each value from the domain into the function and calculating the corresponding output.
Representing the Range
The range can be represented in different ways, such as:
- Set Notation: {15, 11, 5, -3} (as we have done above)
- Listing the values: 15, 11, 5, -3
The most common and precise way to represent the range is using set notation, as it clearly defines the set of all possible output values.
Key Takeaways and Conclusion
In this comprehensive guide, we explored the fundamental concepts of domain and range in functions. We learned that the domain is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). The range is dependent on both the function itself and the specified domain. We demonstrated how to calculate the range for a given domain by substituting each value from the domain into the function and determining the corresponding output. By working through the example of the linear function y = (-2/3)x + 7 with the domain {-12, -6, 3, 15}, we found the range to be {15, 11, 5, -3}.
Importance of Understanding Domain and Range
Understanding domain and range is crucial for several reasons:
- Function Analysis: It helps us understand the behavior of a function and its limitations.
- Graphing: Knowing the domain and range allows us to accurately graph functions.
- Real-World Applications: Many real-world scenarios can be modeled using functions, and understanding their domain and range helps us interpret the results in context.
- Advanced Mathematics: Domain and range are foundational concepts for more advanced topics in mathematics, such as calculus and analysis.
By mastering the concepts of domain and range, you will be well-equipped to tackle a wide range of mathematical problems and gain a deeper understanding of functions and their applications.
This guide has provided you with a solid foundation for understanding domain and range. Practice applying these concepts to different functions and domains to further solidify your knowledge and skills. Remember, understanding functions is key to unlocking the door to more advanced mathematical concepts.
