Finding The Range Of G(x) √(x-1) + 2 A Step-by-Step Guide

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In this comprehensive guide, we will delve deep into understanding the range of the function g(x) = √(x-1) + 2. The range of a function represents the set of all possible output values (y-values) that the function can produce. Determining the range is a fundamental concept in mathematics, particularly in the study of functions and their behavior. We will explore this concept in detail, providing a step-by-step analysis to help you grasp the underlying principles and confidently solve similar problems. By the end of this guide, you will have a solid understanding of how to identify the range of functions involving square roots and transformations. The ability to determine the range of a function is crucial in various mathematical applications, including graphing, calculus, and real-world problem-solving. This guide will equip you with the necessary skills and knowledge to tackle these challenges effectively. Let's embark on this journey of mathematical exploration and unravel the intricacies of function ranges.

H2: Key Concepts Defining the Range of a Function

Before we dive into the specifics of our function, it's essential to grasp the fundamental concepts that define the range of a function. The range is the set of all possible output values (y-values) that a function can produce. It's crucial to distinguish the range from the domain, which represents the set of all possible input values (x-values) that the function can accept. Understanding the interplay between the domain and the function's behavior is key to accurately determining the range. The range can be influenced by several factors, including the type of function (e.g., linear, quadratic, square root), transformations applied to the function (e.g., shifts, stretches, reflections), and any restrictions imposed by the function's definition (e.g., division by zero, square roots of negative numbers). For instance, a linear function with no restrictions will have a range of all real numbers, while a quadratic function will have a range bounded by its vertex. When dealing with square root functions, the non-negativity of the radicand (the expression under the square root) plays a crucial role in determining both the domain and the range. Similarly, transformations such as vertical shifts and stretches will directly impact the range of the function. In the case of g(x) = √(x-1) + 2, we will carefully consider the square root function, the horizontal shift caused by "x-1", and the vertical shift caused by "+2" to accurately determine the range. These concepts will form the foundation for our step-by-step analysis.

H3: Analyzing the Square Root Function and Its Impact on the Range

The square root function is the cornerstone of our analysis, as it dictates the fundamental behavior of g(x) = √(x-1) + 2. The basic square root function, f(x) = √x, has a crucial characteristic: it only produces non-negative output values. This is because the square root of a number is defined as the non-negative value that, when multiplied by itself, yields the original number. Consequently, the range of f(x) = √x is y ≥ 0. This inherent non-negativity of the square root function is the key to understanding why the range of g(x) will be bounded below. Now, let's consider how transformations affect this fundamental range. The function g(x) = √(x-1) + 2 is a transformed version of the basic square root function. The "x-1" inside the square root represents a horizontal shift, which affects the domain but not the range directly. However, the "+2" outside the square root represents a vertical shift, which directly impacts the range. This vertical shift raises the entire graph of the square root function by 2 units. Therefore, the minimum possible y-value of g(x) will be 2, and all other y-values will be greater than or equal to 2. In essence, the range of the square root component, which is non-negative, is then shifted upward by 2 units, leading to the range of the entire function. Understanding this interplay between the basic square root function and the transformations applied to it is crucial for accurately determining the range of functions like g(x).

H3: Understanding Transformations in g(x) = √(x-1) + 2

The function g(x) = √(x-1) + 2 is a prime example of how transformations can alter the behavior and range of a basic function. In this case, we start with the basic square root function, f(x) = √x, and apply two key transformations: a horizontal shift and a vertical shift. The term "x-1" inside the square root represents a horizontal shift of the graph 1 unit to the right. This shift affects the domain of the function, changing it from x ≥ 0 for f(x) to x ≥ 1 for g(x). However, a horizontal shift does not impact the range, as it only moves the graph left or right without altering the possible y-values. The crucial transformation that affects the range is the "+2" outside the square root. This represents a vertical shift of the graph 2 units upward. A vertical shift directly affects the range by adding a constant value to all the y-values of the original function. Since the range of the basic square root function f(x) = √x is y ≥ 0, the vertical shift of 2 units upward transforms this range to y ≥ 2 for g(x) = √(x-1) + 2. In simpler terms, the entire graph of the square root function is lifted 2 units, so the lowest possible y-value becomes 2 instead of 0. By carefully analyzing these transformations, we can see how they shape the graph of the function and, more importantly, how they influence its range. Understanding transformations is a fundamental skill in mathematics, allowing us to predict the behavior of complex functions based on their relationship to simpler, well-known functions.

H2: Determining the Range of g(x) = √(x-1) + 2 Step-by-Step

Now that we've laid the groundwork by understanding the key concepts and the role of transformations, let's proceed with a step-by-step determination of the range of g(x) = √(x-1) + 2. This systematic approach will not only provide the answer but also reinforce the underlying principles involved.

  1. Identify the parent function: The parent function here is the square root function, f(x) = √x. As we discussed earlier, the range of this function is y ≥ 0. This serves as our starting point.

  2. Analyze the transformations: We have two transformations in g(x): a horizontal shift of 1 unit to the right (due to "x-1") and a vertical shift of 2 units upward (due to "+2"). Remember, the horizontal shift affects the domain, not the range. The vertical shift is the key to determining the range of g(x).

  3. Apply the vertical shift to the range: The vertical shift of 2 units upward means that we add 2 to every y-value in the range of the parent function. Since the range of f(x) = √x is y ≥ 0, adding 2 to this inequality gives us y ≥ 0 + 2, which simplifies to y ≥ 2.

  4. State the range: Based on our analysis, the range of g(x) = √(x-1) + 2 is y ≥ 2. This means that the function can produce any y-value greater than or equal to 2.

This step-by-step process clearly demonstrates how we can systematically determine the range of a function by considering the parent function and the transformations applied to it. By understanding these principles, you can confidently tackle similar problems involving various types of functions and transformations.

H2: The Correct Answer and Why It's Correct

Having gone through the detailed analysis, we can now confidently identify the correct answer to the question: What is the range of the function g(x) = √(x-1) + 2?

Based on our step-by-step determination, the correct answer is D. y ≥ 2.

Let's recap why this is the correct answer. The function g(x) = √(x-1) + 2 is a transformation of the basic square root function, f(x) = √x. The square root function inherently produces non-negative values, meaning its range is y ≥ 0. The transformation includes a horizontal shift (which doesn't affect the range) and a crucial vertical shift of 2 units upward. This vertical shift adds 2 to all the y-values of the original function. Therefore, the range of g(x) is obtained by adding 2 to the range of f(x), resulting in y ≥ 0 + 2, which simplifies to y ≥ 2. This confirms that the function g(x) can produce any y-value greater than or equal to 2. The other options are incorrect because they do not account for the vertical shift of 2 units upward. Option A (y ≥ 1) is close but misses the crucial +2 shift. Options B (y ≤ 1) and C (y ≤ 2) suggest that the function's output values are bounded above, which is incorrect for a square root function with a positive vertical shift. By understanding the impact of transformations on the range of the parent function, we can confidently arrive at the correct answer: D. y ≥ 2.

H2: Common Mistakes and How to Avoid Them

When determining the range of functions, especially those involving square roots and transformations, it's easy to make mistakes if you're not careful. Let's explore some common pitfalls and how to avoid them to ensure you arrive at the correct answer consistently.

  1. Ignoring the vertical shift: A frequent mistake is focusing solely on the square root function's inherent non-negativity (y ≥ 0) and overlooking the vertical shift. In the case of g(x) = √(x-1) + 2, the "+2" is crucial. Failing to account for this shift will lead to an incorrect range of y ≥ 0 instead of the correct y ≥ 2. To avoid this, always carefully examine the function for any vertical shifts (or vertical stretches/compressions) and their impact on the y-values.

  2. Confusing domain and range: Another common error is mixing up the domain and range. The domain refers to the possible x-values, while the range refers to the possible y-values. The horizontal shift in g(x) affects the domain (x ≥ 1) but not the range directly. To prevent this, always clearly define what you are looking for – whether it's the set of possible inputs (domain) or the set of possible outputs (range).

  3. Incorrectly interpreting inequalities: Sometimes, the inequality symbol is misinterpreted. For example, y ≥ 2 means y is greater than or equal to 2. Choosing y > 2 would be incorrect as it excludes the value 2 itself. To avoid this, pay close attention to the inequality symbols (≥, ≤, >, <) and what they represent.

  4. Not considering the parent function: Forgetting the fundamental range of the parent function (in this case, f(x) = √x) can lead to errors. Always start by identifying the parent function and its range, then consider the impact of transformations. By being mindful of these common mistakes and actively working to avoid them, you can significantly improve your accuracy in determining the range of functions.

H2: Conclusion Mastering the Art of Range Determination

In conclusion, understanding the range of a function is a fundamental concept in mathematics, crucial for various applications. Throughout this guide, we've meticulously explored the process of determining the range of the function g(x) = √(x-1) + 2. We began by defining the range and distinguishing it from the domain. We then delved into the specifics of the square root function and its inherent non-negativity, which forms the basis for understanding the range of g(x). Furthermore, we analyzed the transformations applied to the basic square root function – the horizontal shift and, most importantly, the vertical shift – and how they influence the range. By following a step-by-step approach, we confidently determined that the range of g(x) = √(x-1) + 2 is y ≥ 2. We also addressed common mistakes to avoid, such as ignoring vertical shifts and confusing domain with range. By mastering the concepts and techniques discussed in this guide, you will be well-equipped to determine the range of a wide variety of functions, not just those involving square roots. The ability to analyze functions, identify transformations, and understand their impact on the range is a valuable skill that will serve you well in your mathematical journey. Remember, practice is key. The more you work with different functions and transformations, the more confident and proficient you will become in determining their ranges. So, continue to explore, experiment, and deepen your understanding of this essential mathematical concept.