Finding The Range Of F(x) = 9x - 17 With Domain {1, 3, 5}

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Finding the range of a function is a fundamental concept in mathematics, particularly in the study of functions and their behavior. The range represents the set of all possible output values (y-values) that a function can produce when given a specific set of input values (x-values), known as the domain. In this comprehensive guide, we will delve deep into the process of determining the range of the linear function f(x) = 9x - 17 for the domain {1, 3, 5}. We will break down the problem step-by-step, explaining the underlying principles and illustrating the concepts with clear examples. Whether you are a student learning about functions for the first time or a seasoned mathematician looking for a refresher, this article aims to provide a thorough understanding of how to find the range of a function.

Defining the Function and Domain

Before we embark on the journey of finding the range, it is crucial to clearly define the function and its domain. The function we are dealing with is a linear function, expressed as f(x) = 9x - 17. This means that for every input value x, the function multiplies it by 9 and then subtracts 17 to produce the output value f(x). The domain, on the other hand, is the set of input values for which the function is defined. In our case, the domain is given as {1, 3, 5}, which means we are only interested in the output values that the function produces when x is 1, 3, or 5. Understanding these basic definitions is the cornerstone for successfully finding the range of the function.

Step-by-Step Calculation of the Range

Now that we have a firm grasp of the function and its domain, we can proceed with the calculation of the range. The process involves evaluating the function f(x) = 9x - 17 for each value in the domain {1, 3, 5}. This means substituting each x-value into the function and calculating the corresponding f(x) value. Let's break it down step-by-step:

  1. For x = 1: f(1) = 9(1) - 17 = 9 - 17 = -8 When the input is 1, the function outputs -8.

  2. For x = 3: f(3) = 9(3) - 17 = 27 - 17 = 10 When the input is 3, the function outputs 10.

  3. For x = 5: f(5) = 9(5) - 17 = 45 - 17 = 28 When the input is 5, the function outputs 28.

By systematically evaluating the function for each value in the domain, we have obtained the corresponding output values. These output values collectively form the range of the function for the given domain.

Identifying the Range

Having calculated the output values for each input in the domain, we can now identify the range. The range is simply the set of all the output values we obtained in the previous step. In this case, the output values are -8, 10, and 28. Therefore, the range of the function f(x) = 9x - 17 for the domain {1, 3, 5} is {-8, 10, 28}. It is important to note that the range is a set, and the order in which the elements are listed does not matter. However, it is customary to list the elements in ascending order for clarity.

Visualizing the Function and Range

To further enhance our understanding, let's visualize the function and its range. The function f(x) = 9x - 17 is a linear function, which means its graph is a straight line. The domain {1, 3, 5} represents three specific points on the x-axis. The range {-8, 10, 28} represents the corresponding y-values on the y-axis. If we were to plot these points on a coordinate plane, we would see three distinct points: (1, -8), (3, 10), and (5, 28). These points lie on the line represented by the function f(x) = 9x - 17. Visualizing the function in this way provides a geometric interpretation of the range and how it relates to the domain.

Key Takeaways and Applications

In this comprehensive exploration, we have successfully determined the range of the function f(x) = 9x - 17 for the domain {1, 3, 5}. We learned that the range is the set of all possible output values that the function can produce for the given domain. We followed a step-by-step approach, evaluating the function for each value in the domain and collecting the resulting output values. We also visualized the function and its range on a coordinate plane, which provided a geometric perspective on the concept. Understanding the range of a function is crucial in various mathematical applications, including graphing functions, solving equations, and analyzing data. It allows us to determine the possible output values of a function and understand its behavior within a specific domain.

Further Exploration and Practice

To solidify your understanding of finding the range of a function, I recommend exploring additional examples and practice problems. You can try different types of functions, such as quadratic functions or trigonometric functions, and different types of domains, such as intervals or sets of real numbers. You can also explore the concept of the range in more advanced mathematical contexts, such as calculus and real analysis. The more you practice, the more comfortable and confident you will become in your ability to determine the range of a function. Remember, mathematics is a journey of exploration and discovery, and there is always more to learn!

Conclusion

In conclusion, finding the range of the function f(x) = 9x - 17 for the domain {1, 3, 5} involves a straightforward process of evaluating the function at each point in the domain and collecting the resulting output values. The range, in this case, is {-8, 10, 28}. This exercise provides a fundamental understanding of how functions map inputs to outputs and how the domain and range define the scope of a function's behavior. Mastering this concept is essential for further studies in mathematics and its applications in various fields. By understanding the relationship between a function, its domain, and its range, we gain a powerful tool for analyzing and interpreting mathematical models of the world around us. This detailed exploration has equipped you with the knowledge and skills necessary to confidently tackle similar problems and delve deeper into the fascinating world of functions.

Functions are a cornerstone of mathematics, providing a framework for understanding relationships between inputs and outputs. A crucial aspect of analyzing a function is determining its range, which is the set of all possible output values. The range is intimately linked to the domain of the function, which defines the set of allowable input values. In this article, we will undertake a detailed exploration of how to find the range of the linear function f(x) = 9x - 17 when the domain is restricted to the set {1, 3, 5}. This example provides a concrete illustration of the fundamental principles involved in determining the range of a function.

Understanding Domain and Range

Before diving into the specific example, it is essential to have a clear understanding of the concepts of domain and range. The domain of a function is the set of all possible input values (often denoted as x) for which the function is defined. In other words, it's the set of values that you can