What Is The Range Of The Function M(x) = √(x-3) + 1? Options: A. (-∞, 1] B. [3, ∞] C. [1, ∞] D. (-∞, 3]

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In mathematics, understanding the range of a function is crucial for grasping its behavior and characteristics. The range of a function represents the set of all possible output values (y-values) that the function can produce. To determine the range, we need to analyze the function's structure, identify any restrictions on the input (domain), and consider how these factors influence the output. In this article, we will explore the function m(x) = √(x-3) + 1 and methodically determine its range. This involves understanding the square root function, its domain, and how transformations affect the output values. Accurately identifying the range is essential for various mathematical applications, including graphing, solving equations, and analyzing real-world phenomena modeled by functions. Our step-by-step approach will clarify the concepts and provide a solid foundation for handling similar problems in the future. The range of a function is a fundamental concept in mathematics, particularly in algebra and calculus, and it provides critical insights into the function's behavior and potential outputs. When analyzing a function, we often focus on what values the function can produce, and this set of values constitutes the range. Understanding the range is not just an academic exercise; it has practical implications in various fields, including physics, engineering, and economics, where functions are used to model real-world phenomena. In this detailed exploration, we aim to provide a clear and comprehensive understanding of how to determine the range of a function, specifically focusing on the example function m(x) = √(x-3) + 1. By the end of this discussion, you will have a robust understanding of the principles involved and be able to apply them to other functions as well. This mastery will not only aid in academic pursuits but also in practical problem-solving scenarios where understanding function behavior is critical.

Analyzing the Function m(x) = √(x-3) + 1

Let’s start by dissecting the function m(x) = √(x-3) + 1. This function involves a square root, which introduces certain constraints. The square root function, denoted as √x, is only defined for non-negative values. This means that the expression inside the square root must be greater than or equal to zero. In our case, the expression inside the square root is (x-3). Therefore, we have the inequality x - 3 ≥ 0. Solving this inequality gives us x ≥ 3. This tells us that the domain of the function m(x) is all real numbers greater than or equal to 3. The domain is a crucial piece of information because it directly affects the possible output values, which constitute the range. The square root function itself, √x, produces non-negative values. The smallest value it can produce is 0, which occurs when x = 0. As x increases, √x also increases, but it never becomes negative. This characteristic of the square root function is vital in determining the range of our given function. The transformation √(x-3) shifts the standard square root function 3 units to the right along the x-axis. This shift affects the domain but doesn't change the fundamental non-negative nature of the output. The key point here is that √(x-3) will still produce values greater than or equal to 0. The “+ 1” in the function m(x) = √(x-3) + 1 represents a vertical shift. Adding 1 to the entire function means that every output value of √(x-3) is increased by 1. This shift directly impacts the range of the function. Understanding how each component of the function affects the overall output is critical for accurately determining the range. We’ve established that the square root portion, √(x-3), produces non-negative values, and then we add 1 to those values. This sets the stage for determining the minimum output value and, consequently, the range of the function. Understanding the behavior of the square root function and the impact of transformations such as horizontal shifts and vertical shifts is foundational for analyzing the range of functions like m(x) = √(x-3) + 1. The domain restriction imposed by the square root, combined with the vertical shift, will lead us to the correct range.

Determining the Range of m(x)

Now, let's methodically determine the range of m(x) = √(x-3) + 1. We know that the expression inside the square root, x - 3, must be non-negative. Thus, the smallest value that √(x-3) can take is 0, which occurs when x = 3. When x = 3, m(3) = √(3-3) + 1 = √0 + 1 = 0 + 1 = 1. This tells us that the smallest possible value of m(x) is 1. Since the square root function √(x-3) can only produce non-negative values, and we are adding 1 to it, the output of m(x) will always be greater than or equal to 1. As x increases beyond 3, the value of √(x-3) also increases. Therefore, m(x) will increase without bound as x increases. There is no upper limit to the output values of m(x). The function m(x) starts at 1 and increases indefinitely, which means the range includes all real numbers greater than or equal to 1. Mathematically, we represent this as [1, ∞). This notation indicates that the range includes 1 and extends to positive infinity. The bracket “[“ signifies that 1 is included in the range, while the parenthesis “(“ next to infinity signifies that infinity is not a specific number but rather a concept of unboundedness. To solidify our understanding, let's consider a few more points. For example, if x = 4, m(4) = √(4-3) + 1 = √1 + 1 = 1 + 1 = 2. If x = 7, m(7) = √(7-3) + 1 = √4 + 1 = 2 + 1 = 3. As we choose larger values of x, m(x) continues to increase, confirming that the range extends to infinity. By understanding the behavior of the square root function and the impact of the vertical shift, we have confidently determined that the range of m(x) = √(x-3) + 1 is [1, ∞). This detailed analysis highlights the importance of considering both the domain and the transformations applied to a function when determining its range. The range [1, ∞) accurately describes the set of all possible output values for the function m(x), providing a complete picture of its behavior. Our methodical approach ensures that we have accounted for all relevant factors, leading us to the correct conclusion.

The Correct Answer: C. [1, ∞)

Based on our comprehensive analysis, the correct answer for the range of the function m(x) = √(x-3) + 1 is C. [1, ∞). We methodically examined the function, considered the domain restriction imposed by the square root, and analyzed the impact of the vertical shift. This step-by-step approach allowed us to confidently determine that the smallest value of m(x) is 1 and that the function increases without bound as x increases. Option A, (-∞, 1], is incorrect because it implies that the function can take on negative values, which is not possible due to the square root and the vertical shift of +1. The square root function always produces non-negative values, and adding 1 ensures that the output is always greater than or equal to 1. Option B, [3, ∞), is incorrect because it confuses the domain with the range. The interval [3, ∞) represents the domain of the function, i.e., the set of all possible input values for x. The range, on the other hand, represents the set of all possible output values, which are different. Option D, (-∞, 3], is also incorrect for the same reason as Option A. It suggests that the function can produce values less than 1, which is not the case. The vertical shift ensures that the output values start at 1 and increase from there. Our analysis has clearly demonstrated that the range of m(x) includes all real numbers greater than or equal to 1, which is precisely what the interval [1, ∞) represents. This thorough examination of each option reinforces the importance of a systematic approach to solving mathematical problems. Understanding the fundamental properties of functions, such as the square root, and how transformations affect them is crucial for accurately determining the range. By considering the domain, analyzing the transformations, and evaluating the function's behavior, we arrived at the correct answer, [1, ∞). This careful and detailed analysis not only provides the correct answer but also solidifies our understanding of function ranges.

In conclusion, determining the range of a function like m(x) = √(x-3) + 1 requires a solid understanding of the function’s components, domain restrictions, and transformations. We started by recognizing the impact of the square root, which necessitates a non-negative expression inside. This led us to establish the domain as x ≥ 3. We then analyzed the vertical shift of +1, which raised the minimum output value. By systematically considering these factors, we accurately identified the range as [1, ∞). This process underscores the importance of a methodical approach in mathematics. Each step, from understanding the domain to analyzing transformations, contributes to a comprehensive understanding of the function's behavior. This understanding is not only crucial for solving specific problems but also for developing a broader mathematical intuition. The ability to determine the range of a function is a fundamental skill in mathematics, with applications across various fields, including calculus, algebra, and real-world modeling. By mastering these concepts, you can tackle more complex mathematical challenges with confidence. The range of a function provides valuable insights into its behavior, allowing us to understand the possible output values and make informed predictions. This knowledge is essential for graphing functions, solving equations, and applying mathematical models to real-world scenarios. The process of determining the range involves a combination of algebraic techniques, analytical thinking, and a deep understanding of function properties. By consistently applying these principles, you can build a strong foundation in mathematics and enhance your problem-solving skills. We encourage you to practice with different functions and variations to further solidify your understanding. Exploring various types of functions, such as quadratics, exponentials, and trigonometric functions, will broaden your mathematical toolkit and enable you to analyze a wider range of problems. Remember, mathematics is a journey of learning and discovery, and each problem you solve contributes to your overall understanding and expertise.