Square Root Function Transformation Domain [2, ∞) And Range [3, ∞)

The core concept of square root function transformations revolves around manipulating the parent square root function, often denoted as f(x) = √x, to achieve desired domain and range characteristics. The domain of a function represents the set of all possible input values (x-values), while the range encompasses the set of all possible output values (y-values). Transformations, such as shifts and reflections, directly impact these domain and range attributes.

Understanding the parent square root function's domain and range provides a crucial foundation. The parent function, f(x) = √x, has a domain of [0, ∞) and a range of [0, ∞). This signifies that the function accepts non-negative input values and produces non-negative output values. Transformations serve as tools to alter these fundamental characteristics, enabling us to tailor the function to specific requirements.

Horizontal Translations

Horizontal translations involve shifting the parent square root function horizontally along the x-axis. This transformation is achieved by adding or subtracting a constant within the square root expression. For instance, the function g(x) = √(x - h) represents a horizontal shift of the parent function by h units. A positive value of h corresponds to a shift to the right, while a negative value signifies a shift to the left. These shifts directly affect the domain of the transformed function. Shifting the parent function to the right by h units results in a domain of [h, ∞), while a shift to the left by h units yields a domain of [-h, ∞).

To illustrate, consider the function g(x) = √(x - 2). This function represents a horizontal shift of the parent function f(x) = √x by 2 units to the right. Consequently, the domain of g(x) becomes [2, ∞). This means that the function accepts input values greater than or equal to 2. Conversely, the function j(x) = √(x + 2) represents a horizontal shift of the parent function by 2 units to the left, resulting in a domain of [-2, ∞).

Vertical Translations

Vertical translations involve shifting the parent square root function vertically along the y-axis. This transformation is accomplished by adding or subtracting a constant outside the square root expression. The function g(x) = √x + k represents a vertical shift of the parent function by k units. A positive value of k corresponds to an upward shift, while a negative value signifies a downward shift. These shifts directly influence the range of the transformed function. Shifting the parent function upward by k units results in a range of [k, ∞), while a shift downward by k units yields a range of [-k, ∞).

Consider the function g(x) = √x + 3. This function represents a vertical shift of the parent function f(x) = √x by 3 units upward. Consequently, the range of g(x) becomes [3, ∞). This means that the function produces output values greater than or equal to 3. Conversely, the function j(x) = √x - 3 represents a vertical shift of the parent function by 3 units downward, resulting in a range of [-3, ∞).

Combining Horizontal and Vertical Translations

In many instances, horizontal and vertical translations are combined to achieve specific domain and range requirements. The general form of a square root function incorporating both horizontal and vertical shifts is g(x) = √(x - h) + k, where h represents the horizontal shift and k represents the vertical shift. The domain of this transformed function is [h, ∞), and the range is [k, ∞). Understanding how these shifts interact is crucial for accurately predicting the resulting domain and range.

For instance, the function g(x) = √(x - 2) + 3 represents a horizontal shift of the parent function f(x) = √x by 2 units to the right and a vertical shift by 3 units upward. Consequently, the domain of g(x) is [2, ∞), and the range is [3, ∞). This combination of shifts allows us to precisely control the positioning of the square root function on the coordinate plane.

Determining the Correct Transformation

When presented with a specific domain and range, determining the correct transformation involves analyzing the required shifts. The domain provides insights into the horizontal shift, while the range reveals the vertical shift. By carefully examining these characteristics, we can deduce the appropriate values of h and k in the general form g(x) = √(x - h) + k.

Consider the problem of finding the transformation of the parent square root function that results in a domain of [2, ∞) and a range of [3, ∞). The domain [2, ∞) indicates a horizontal shift of 2 units to the right, suggesting that h = 2. The range [3, ∞) implies a vertical shift of 3 units upward, indicating that k = 3. Therefore, the correct transformation is g(x) = √(x - 2) + 3.

Analyzing the Given Options

In a multiple-choice scenario, we are typically presented with several options, each representing a different transformation. To identify the correct answer, we must analyze each option and determine its corresponding domain and range. This involves applying the principles of horizontal and vertical shifts discussed earlier.

Let's examine the options provided in the original prompt:

A. g(x) = √(x - 2) + 3

B. j(x) = √(x + 2) + 3

Option A, g(x) = √(x - 2) + 3, represents a horizontal shift of 2 units to the right and a vertical shift of 3 units upward. This results in a domain of [2, ∞) and a range of [3, ∞), matching the given requirements. Therefore, option A is the correct answer.

Option B, j(x) = √(x + 2) + 3, represents a horizontal shift of 2 units to the left and a vertical shift of 3 units upward. This results in a domain of [-2, ∞) and a range of [3, ∞), which does not match the given domain of [2, ∞). Therefore, option B is incorrect.

Conclusion

Mastering square root function transformations requires a solid understanding of horizontal and vertical shifts and their impact on the domain and range. By analyzing the given domain and range, we can effectively determine the required transformations and identify the correct function. This knowledge is essential for solving a wide range of mathematical problems involving square root functions.

Let's delve into the transformations of the square root function and how they affect the domain and range. Understanding these transformations is crucial for manipulating functions to fit specific criteria. Our primary focus is identifying the correct transformation of the parent square root function that results in a domain of $[2, \infty)$ and a range of $[3, \infty)$.

Understanding the Parent Square Root Function

The parent square root function, denoted as $f(x) = \sqrt{x}$, serves as the foundation for all transformations. This function has a natural domain of $[0, \infty)$, meaning it accepts only non-negative real numbers as input, and a range of $[0, \infty)$, indicating that its output is also non-negative. Transforming this basic function involves shifting it horizontally or vertically, reflecting it, or stretching/compressing it. We'll focus on translations, which involve shifting the function without altering its shape.

Transformations and Their Effects

Transformations can be broadly categorized into horizontal and vertical shifts. A horizontal shift involves adding or subtracting a constant inside the square root, affecting the domain. A vertical shift involves adding or subtracting a constant outside the square root, impacting the range. Let's dissect these transformations to understand their individual effects and then see how they combine to meet our domain and range requirements.

Horizontal Shifts

A horizontal shift is achieved by modifying the input $x$ inside the square root. Consider a function of the form $g(x) = \sqrt{x - h}$. If $h$ is positive, the graph shifts to the right by $h$ units. If $h$ is negative, the graph shifts to the left by $|h|$ units. This shift directly affects the domain of the function. For $g(x) = \sqrt{x - h}$, the domain becomes $[h, \infty)$. This is because the expression inside the square root, $x - h$, must be non-negative, leading to $x - h \geq 0$, and thus $x \geq h$.

For example, if we have $g(x) = \sqrt{x - 2}$, the graph shifts 2 units to the right, and the domain becomes $[2, \infty)$. This means that the function is defined for all real numbers greater than or equal to 2. On the other hand, if we have $g(x) = \sqrt{x + 2}$, which can be rewritten as $g(x) = \sqrt{x - (-2)}$, the graph shifts 2 units to the left, and the domain becomes $[-2, \infty)$.

Vertical Shifts

A vertical shift is accomplished by adding or subtracting a constant outside the square root. Consider a function of the form $g(x) = \sqrt{x} + k$. If $k$ is positive, the graph shifts upward by $k$ units. If $k$ is negative, the graph shifts downward by $|k|$ units. This shift directly affects the range of the function. For $g(x) = \sqrt{x} + k$, the range becomes $[k, \infty)$. This is because the square root part, $\sqrt{x}$, is always non-negative, and adding $k$ to it shifts the entire range by $k$ units.

For instance, if we have $g(x) = \sqrt{x} + 3$, the graph shifts 3 units upward, and the range becomes $[3, \infty)$. This means that the output of the function is always greater than or equal to 3. Conversely, if we have $g(x) = \sqrt{x} - 3$, the graph shifts 3 units downward, and the range becomes $[-3, \infty)$.

Combining Horizontal and Vertical Shifts

The power of transformations lies in combining them. We can shift the square root function both horizontally and vertically by using a function of the form $g(x) = \sqrt{x - h} + k$. In this case, the graph shifts $h$ units horizontally and $k$ units vertically. The domain becomes $[h, \infty)$, and the range becomes $[k, \infty)$. This combination allows us to precisely position the square root function in the coordinate plane.

Determining the Correct Transformation for the Given Domain and Range

Our task is to find the transformation that results in a domain of $[2, \infty)$ and a range of $[3, \infty)$. To achieve this, we need to identify the appropriate horizontal and vertical shifts. The domain $[2, \infty)$ indicates that the graph has been shifted 2 units to the right, implying that $h = 2$. The range $[3, \infty)$ suggests that the graph has been shifted 3 units upward, meaning that $k = 3$. Therefore, the correct transformation is $g(x) = \sqrt{x - 2} + 3$.

This function incorporates both a horizontal shift of 2 units to the right and a vertical shift of 3 units upward. The expression inside the square root, $x - 2$, ensures that the domain is $[2, \infty)$, as $x - 2 \geq 0$ implies $x \geq 2$. The addition of 3 outside the square root guarantees that the range is $[3, \infty)$, since $\sqrt{x - 2}$ is always non-negative, and adding 3 shifts the entire range up by 3 units.

Analyzing the Options

Let's analyze the given options to confirm our conclusion:

A. $g(x) = \sqrt{x - 2} + 3$

This function represents a horizontal shift of 2 units to the right (due to $x - 2$) and a vertical shift of 3 units upward (due to $+3$). The domain is indeed $[2, \infty)$, and the range is $[3, \infty)$. Thus, this option matches the required domain and range.

B. $j(x) = \sqrt{x + 2} + 3$

This function represents a horizontal shift of 2 units to the left (due to $x + 2$) and a vertical shift of 3 units upward (due to $+3$). The domain is $[-2, \infty)$, and the range is $[3, \infty)$. While the range matches, the domain does not, as it should be $[2, \infty)$.

Conclusion

Based on our analysis, the correct transformation of the parent square root function that results in a domain of $[2, \infty)$ and a range of $[3, \infty)$ is $g(x) = \sqrt{x - 2} + 3$. This function embodies both the necessary horizontal and vertical shifts to meet the specified criteria. Understanding these transformations allows us to manipulate functions and tailor them to specific requirements, a fundamental skill in mathematics. In summary, the square root function $g(x) = \sqrt{x - 2} + 3$ perfectly fits the domain and range requirements, showcasing the power of transformations in function manipulation.

In this discussion, we aim to identify the correct transformation of the parent square root function that yields a domain of $[2, \infty)$ and a range of $[3, \infty)$. To achieve this, we must thoroughly understand the impact of various transformations on the function's domain and range. The transformations we'll focus on are horizontal and vertical translations, which shift the graph of the function without altering its fundamental shape.

Understanding the Parent Square Root Function and Its Properties

The parent square root function, commonly expressed as $f(x) = \sqrt{x}$, serves as the foundational element for our analysis. This function is characterized by its domain and range, both of which are $[0, \infty)$. This means that the function accepts only non-negative real numbers as input (domain) and produces only non-negative real numbers as output (range). The graph of the parent function starts at the origin (0, 0) and extends to the right, gradually increasing in value. Any transformation applied to this parent function will alter its position and, consequently, its domain and range. Let’s break down the types of transformations that can occur.

Horizontal Translations: Shifting Along the x-axis

Horizontal translations involve shifting the graph of the function left or right along the x-axis. This transformation is achieved by adding or subtracting a constant within the square root expression. The general form of a horizontally translated square root function is $g(x) = \sqrt{x - h}$, where $h$ represents the horizontal shift. If $h$ is positive, the graph shifts to the right by $h$ units, while if $h$ is negative, the graph shifts to the left by $|h|$ units. The domain of the transformed function becomes $[h, \infty)$, as the expression inside the square root ($x - h$) must be non-negative, i.e., $x - h \geq 0$, which implies $x \geq h$.

Consider the function $g(x) = \sqrt{x - 2}$. This represents a horizontal shift of the parent function 2 units to the right. Consequently, the domain of $g(x)$ is $[2, \infty)$, as the function is only defined for $x$ values greater than or equal to 2. Conversely, the function $g(x) = \sqrt{x + 2}$ (which can be written as $g(x) = \sqrt{x - (-2)}$) represents a horizontal shift of 2 units to the left, resulting in a domain of $[-2, \infty)$.

Vertical Translations: Shifting Along the y-axis

Vertical translations involve shifting the graph of the function up or down along the y-axis. This transformation is achieved by adding or subtracting a constant outside the square root expression. The general form of a vertically translated square root function is $g(x) = \sqrt{x} + k$, where $k$ represents the vertical shift. If $k$ is positive, the graph shifts upward by $k$ units, while if $k$ is negative, the graph shifts downward by $|k|$ units. The range of the transformed function becomes $[k, \infty)$, as the square root part ($\sqrt{x}$) is always non-negative, and adding $k$ to it shifts the entire range by $k$ units.

For example, the function $g(x) = \sqrt{x} + 3$ represents a vertical shift of the parent function 3 units upward. As a result, the range of $g(x)$ is $[3, \infty)$, meaning the function's output is always greater than or equal to 3. Conversely, the function $g(x) = \sqrt{x} - 3$ represents a vertical shift of 3 units downward, leading to a range of $[-3, \infty)$.

Combining Horizontal and Vertical Translations for Specific Domain and Range

The most effective way to manipulate a function's position is by combining horizontal and vertical translations. The general form of a square root function incorporating both types of shifts is $g(x) = \sqrt{x - h} + k$, where $h$ is the horizontal shift and $k$ is the vertical shift. The domain of this transformed function is $[h, \infty)$, and the range is $[k, \infty)$. This combined approach allows for precise control over the function's location in the coordinate plane.

Determining the Correct Transformation for the Given Requirements

Our primary objective is to identify the transformation that results in a domain of $[2, \infty)$ and a range of $[3, \infty)$. To achieve this, we need to analyze the required shifts based on the given domain and range. The domain $[2, \infty)$ indicates a horizontal shift of 2 units to the right, implying that $h = 2$. The range $[3, \infty)$ suggests a vertical shift of 3 units upward, indicating that $k = 3$. Therefore, the correct transformation should be of the form $g(x) = \sqrt{x - 2} + 3$.

Analyzing the Provided Options

Let's examine the options given in the original question to verify our conclusion:

A. $g(x) = \sqrt{x - 2} + 3$

This function represents a horizontal shift of 2 units to the right and a vertical shift of 3 units upward. The domain, determined by the expression inside the square root, is $[2, \infty)$, as $x - 2 \geq 0$ implies $x \geq 2$. The range is $[3, \infty)$, as the square root part is always non-negative, and adding 3 shifts the range upward by 3 units. Therefore, this option perfectly matches the required domain and range.

B. $j(x) = \sqrt{x + 2} + 3$

This function represents a horizontal shift of 2 units to the left and a vertical shift of 3 units upward. The domain is $[-2, \infty)$, as $x + 2 \geq 0$ implies $x \geq -2$. The range is $[3, \infty)$. While the range matches, the domain does not, making this option incorrect.

Conclusion: The Correct Transformation Identified

In summary, the correct transformation of the parent square root function that yields a domain of $[2, \infty)$ and a range of $[3, \infty)$ is $g(x) = \sqrt{x - 2} + 3$. This function effectively combines a horizontal shift of 2 units to the right and a vertical shift of 3 units upward, perfectly aligning with the specified domain and range requirements. A deep understanding of these transformations enables us to manipulate functions and adapt them to specific criteria, a crucial skill in mathematical problem-solving. The square root function $g(x) = \sqrt{x - 2} + 3$ is the precise answer, demonstrating the power of transformations in shaping function behavior.