Understanding Transformations Range Of Function F(x) = √x

In mathematics, understanding the behavior of functions and how they transform is a fundamental concept. Functions are the building blocks of mathematical models, and transformations allow us to manipulate these functions to fit various scenarios. In this comprehensive exploration, we will delve into the function f(x) = √x, its properties, and how it transforms under a given rule. We will also determine the range of the transformed function, providing a clear understanding of the concepts involved. Let's embark on this mathematical journey to unravel the intricacies of function transformations.

At its core, the function f(x) = √x represents the square root of a given input x. It's a simple yet powerful function that forms the basis for many mathematical concepts. To truly appreciate its significance, let's first dissect its fundamental characteristics. The domain of f(x) = √x is the set of all non-negative real numbers, which can be represented as x ≥ 0. This is because the square root of a negative number is not defined within the realm of real numbers. Similarly, the range of f(x) = √x is also the set of all non-negative real numbers, denoted as y ≥ 0. This is because the square root of a non-negative number is always non-negative. With these foundational properties in mind, we can now delve into the exciting world of transformations.

Function transformations are like the artistic tools of mathematics, allowing us to manipulate and reshape functions in various ways. These transformations can involve shifts, stretches, compressions, and reflections, each with its unique effect on the function's graph. In this particular scenario, we encounter a transformation rule that dictates how the function f(x) = √x is altered. The rule (x, y) → (x - 6, y + 9) describes a transformation that shifts the graph of the function horizontally and vertically. To fully grasp the impact of this transformation, let's dissect it into its components. The (x - 6) part of the rule indicates a horizontal shift of 6 units to the right. This means that every point on the original graph of f(x) = √x is moved 6 units in the positive x-direction. Conversely, the (y + 9) part of the rule signifies a vertical shift of 9 units upward. Consequently, each point on the original graph is also moved 9 units in the positive y-direction. By applying these transformations, we obtain a new function, which we'll call A(x), that is a shifted version of the original f(x) = √x. Now that we have a solid understanding of the transformation rule, let's turn our attention to determining the range of the transformed function A(x).

To determine the range of the transformed function A(x), we need to carefully consider how the transformation affects the original range of f(x) = √x. As we established earlier, the range of f(x) = √x is y ≥ 0, meaning that the function's output values are always non-negative. The vertical shift in the transformation rule plays a crucial role in altering the range. The rule (x, y) → (x - 6, y + 9) includes a vertical shift of +9 units. This means that every y-value in the original function f(x) = √x is increased by 9 to obtain the corresponding y-value in the transformed function A(x). Consequently, the range of A(x) is shifted upward by 9 units. Since the original range of f(x) = √x is y ≥ 0, the range of A(x) becomes y ≥ 0 + 9, which simplifies to y ≥ 9. Therefore, the expression that accurately describes the range of A(x) is y ≥ 9. This signifies that the output values of the transformed function A(x) are always greater than or equal to 9. In essence, the vertical shift in the transformation rule directly impacts the range of the function, causing it to shift accordingly. With a clear understanding of the transformation and its effect on the range, we can confidently conclude that the correct expression for the range of A(x) is y ≥ 9. This exercise exemplifies the importance of understanding function transformations and their impact on key properties like domain and range.

In conclusion, we have embarked on a comprehensive exploration of the function f(x) = √x and its transformation under the rule (x, y) → (x - 6, y + 9). We began by dissecting the fundamental properties of f(x) = √x, including its domain and range. We then delved into the intricacies of function transformations, recognizing that the given rule represents a horizontal shift of 6 units to the right and a vertical shift of 9 units upward. By carefully analyzing the effect of the vertical shift on the range, we determined that the range of the transformed function A(x) is y ≥ 9. This signifies that the output values of A(x) are always greater than or equal to 9, a direct consequence of the vertical shift in the transformation rule. This exploration underscores the significance of understanding function transformations and their impact on key properties like domain and range. By grasping these concepts, we can effectively manipulate and analyze functions to model various mathematical scenarios. The journey through function transformations not only enhances our mathematical acumen but also equips us with the tools to tackle more complex mathematical challenges. As we conclude this exploration, we carry with us a deeper appreciation for the beauty and power of mathematical transformations.

Therefore, the correct answer is not among the options provided in the original question. The accurate expression describing the range of A(x) is y ≥ 9.