Range Of G(x) = √(x-1) + 2 A Step-by-Step Solution

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In the realm of mathematical functions, understanding the range is crucial for comprehending the behavior and output of a function. The range of a function represents the set of all possible output values (y-values) that the function can produce. This article delves into determining the range of the function $g(x) = \sqrt{x-1} + 2$, providing a comprehensive explanation to help you grasp the concept and arrive at the correct answer.

Analyzing the Function g(x) = √(x-1) + 2

To determine the range of the function, let's break down its components and analyze how each part contributes to the overall output. Our function, $g(x) = \sqrt{x-1} + 2$, consists of two main parts: a square root term and a constant term. Understanding the properties of these terms is key to finding the range.

1. The Square Root Term: √(x-1)

The square root function, denoted by $\sqrt }$, has a fundamental characteristic it only produces non-negative values. This means the output of $\sqrt{x-1$ will always be greater than or equal to zero. This is because the square root of a number is defined as a value that, when multiplied by itself, yields the original number. Since squaring any real number (positive, negative, or zero) results in a non-negative value, the square root function is designed to only return non-negative results. This non-negativity is a crucial piece of the puzzle when determining the range of our function.

Now, let's consider the expression inside the square root, which is x-1. For the square root to be defined in the real number system, the expression x-1 must be greater than or equal to zero. This is because we cannot take the square root of a negative number within the realm of real numbers. Therefore, we have the inequality:

x−1≥0x - 1 \geq 0

Solving for x, we get:

x≥1x \geq 1

This inequality tells us that the domain of the function $g(x)$ is all x values greater than or equal to 1. In other words, we can only input values of x that are 1 or larger into the function. This restriction on the input values will also influence the possible output values, which is directly related to the range.

Since the square root of x-1 is always non-negative, the smallest possible value for $\sqrt{x-1}$ is 0. This occurs when x = 1, because $\sqrt{1-1} = \sqrt{0} = 0$. As x increases beyond 1, the value of $\sqrt{x-1}$ also increases. However, it's important to remember that the square root term will never be negative.

2. The Constant Term: +2

The constant term, +2, plays a significant role in shifting the output values of the function. It acts as a vertical shift, moving the entire graph of the function upwards by 2 units. This means that whatever value we get from the square root term, we will always add 2 to it. This constant shift directly affects the range of the function.

Determining the Range of g(x)

Now that we've analyzed the individual components of the function, we can combine our understanding to determine the range of $g(x) = \sqrt{x-1} + 2$. We know that the square root term, $\sqrt{x-1}$, is always greater than or equal to 0. This means the smallest possible value for $\sqrt{x-1}$ is 0.

When $\sqrt{x-1} = 0$, the function becomes:

g(x)=0+2=2g(x) = 0 + 2 = 2

This tells us that the smallest possible output value (y-value) of the function is 2. Since the square root term can only increase as x increases (for x ≥ 1), the output of the function will always be greater than or equal to 2. There is no upper bound on the output, as the square root term can grow infinitely large as x increases. Therefore, the range of the function includes all y-values greater than or equal to 2.

Expressing the Range

We can express the range of the function $g(x) = \sqrt{x-1} + 2$ using different notations:

  • Inequality Notation: $y \geq 2$
  • Interval Notation: $[2, \infty)$

Both notations convey the same information: the range of the function consists of all real numbers y that are greater than or equal to 2. The inequality notation directly states this condition, while the interval notation uses a bracket to indicate that 2 is included in the range and the infinity symbol to represent that there is no upper bound.

The Correct Answer

Based on our analysis, the correct answer to the question "What is the range of function $g(x) = \sqrt{x-1} + 2$?" is:

C. y ≥ 2

This confirms that the function's output values are always greater than or equal to 2, aligning with our understanding of the square root function and the constant term's effect.

Visualizing the Range

To further solidify your understanding, it can be helpful to visualize the function's graph. The graph of $g(x) = \sqrt{x-1} + 2$ starts at the point (1, 2) and extends upwards and to the right. The y-values on the graph represent the output of the function, and you'll notice that they all fall above or on the line y = 2. This graphical representation provides a visual confirmation of the range we determined analytically.

Key Takeaways

  • The range of a function is the set of all possible output values (y-values).
  • The square root function, $\sqrt{ }$, produces non-negative values.
  • The expression inside the square root must be greater than or equal to zero for the function to be defined in the real number system.
  • Constant terms in a function cause vertical shifts in the graph and affect the range.
  • The range of $g(x) = \sqrt{x-1} + 2$ is $y \geq 2$.

Practice Problems

To reinforce your understanding of finding the range of functions, try these practice problems:

  1. Determine the range of the function $f(x) = \sqrt{x+3} - 1$.
  2. What is the range of $h(x) = 2\sqrt{x} + 5$?
  3. Find the range of $k(x) = -\sqrt{x-2} + 4$.

By working through these problems, you'll develop your skills in analyzing functions and identifying their ranges. Remember to consider the properties of the individual components of the function, such as square roots, constants, and transformations.

Conclusion

Determining the range of a function is a fundamental skill in mathematics. By understanding the properties of different function types and their components, you can effectively analyze and identify the set of all possible output values. In the case of $g(x) = \sqrt{x-1} + 2$, we've shown that the range is $y \geq 2$, a result of the non-negative nature of the square root function and the vertical shift caused by the constant term. With a solid grasp of these concepts, you'll be well-equipped to tackle a wide range of problems involving function ranges.

Determining the range of a function is a core concept in mathematics, crucial for understanding the function's behavior and output. This article provides a detailed guide on how to find the range of the function $g(x) = \sqrt{x-1} + 2$. We will explore the components of the function, analyze their impact on the output, and ultimately determine the correct range. The range, in simple terms, encompasses all possible output values (y-values) that the function can produce. By the end of this guide, you'll have a strong understanding of how to approach such problems.

Deconstructing the Function g(x) = √(x-1) + 2

To effectively determine the range, we must first break down the function $g(x) = \sqrt{x-1} + 2$ into its fundamental parts. The function comprises a square root term and a constant term. Each term has its own characteristics and influences the overall output of the function. Let's examine each part in detail to gain a complete understanding of how they interact to define the function's range.

Analyzing the Square Root Component: √(x-1)

The square root function, represented by $\sqrt }$, is the first key element to consider. A crucial property of the square root function is that it produces only non-negative values. This means the output of $\sqrt{x-1}$ will always be greater than or equal to zero. This is inherent to the definition of the square root it returns the non-negative value that, when multiplied by itself, equals the number inside the root. For example, $\sqrt{9$ is 3, not -3, even though (-3) * (-3) also equals 9. This non-negativity characteristic is essential when determining the range of our function.

Next, we need to consider the expression inside the square root, which is x-1. For the square root to be defined within the real number system, the expression x-1 must be greater than or equal to zero. This is because we cannot take the square root of a negative number and obtain a real number result. This constraint gives us an important condition:

x−1≥0x - 1 \geq 0

Solving this inequality for x, we get:

x≥1x \geq 1

This inequality tells us that the domain of the function $g(x)$ is all x values greater than or equal to 1. In other words, we can only input values of x that are 1 or larger into the function. This restriction on the input values significantly influences the possible output values, directly affecting the range.

Since the square root of x-1 is always non-negative, the smallest possible value for $\sqrt{x-1}$ is 0. This occurs when x = 1, because $\sqrt{1-1} = \sqrt{0} = 0$. As x increases beyond 1, the value of $\sqrt{x-1}$ also increases. However, it's crucial to remember that the square root term will never be negative, which limits the lower bound of the range.

Examining the Constant Term: +2

The constant term, +2, plays a vital role in vertically shifting the function's output. This term adds 2 to the value obtained from the square root component. In graphical terms, this shifts the entire function's graph upwards by 2 units. Understanding this vertical shift is crucial for accurately determining the range of the function. The constant term directly impacts the minimum possible output value of the function, effectively setting a lower limit on the range.

Putting It Together: Determining the Range of g(x)

Having analyzed the individual components, we can now combine our understanding to determine the range of $g(x) = \sqrt{x-1} + 2$. We know that the square root term, $\sqrt{x-1}$, is always greater than or equal to 0. This means the smallest possible value for $\sqrt{x-1}$ is 0.

When $\sqrt{x-1} = 0$, the function becomes:

g(x)=0+2=2g(x) = 0 + 2 = 2

This calculation reveals a critical piece of information: the smallest possible output value (y-value) of the function is 2. Since the square root term can only increase as x increases (for x ≥ 1), the output of the function will always be greater than or equal to 2. As x becomes larger, the value of $\sqrt{x-1}$ also increases, leading to larger output values for $g(x)$. There is no upper limit on how large the output can become, meaning the range extends infinitely upwards. Therefore, the range of the function includes all y-values greater than or equal to 2.

Expressing the Range in Different Notations

We can express the range of the function $g(x) = \sqrt{x-1} + 2$ using several notations, each conveying the same information in a slightly different way:

  • Inequality Notation: $y \geq 2$
  • Interval Notation: $[2, \infty)$

Both notations express that the range of the function includes all real numbers y that are greater than or equal to 2. The inequality notation directly states this condition, while the interval notation uses a bracket to indicate that 2 is included in the range and the infinity symbol to represent that there is no upper bound.

Identifying the Correct Answer

Based on our comprehensive analysis, the correct answer to the question "What is the range of function $g(x) = \sqrt{x-1} + 2$?" is:

C. y ≥ 2

This conclusion aligns perfectly with our step-by-step examination of the function's components and their impact on the output values. The square root term, being non-negative, sets a lower bound, and the constant term shifts the entire function upwards, further solidifying the range.

Visual Representation of the Range

To enhance understanding, visualizing the function's graph can be incredibly beneficial. The graph of $g(x) = \sqrt{x-1} + 2$ starts at the point (1, 2) and extends upwards and to the right. The y-values on the graph represent the output of the function, and you'll clearly see that they all fall above or on the line y = 2. This visual representation provides a concrete confirmation of the range we determined through analytical methods.

Key Concepts and Takeaways

  • The range of a function represents all possible output values (y-values).
  • The square root function ($\sqrt{ }$) produces only non-negative values.
  • The expression inside the square root must be greater than or equal to zero for the function to be defined in the real number system.
  • Constant terms in a function cause vertical shifts in the graph, influencing the range.
  • The range of $g(x) = \sqrt{x-1} + 2$ is $y \geq 2$, meaning the output values are always greater than or equal to 2.

Practice Exercises for Mastery

To solidify your understanding of determining the range of functions, try working through these practice exercises:

  1. Determine the range of the function $f(x) = \sqrt{x+3} - 1$.
  2. What is the range of $h(x) = 2\sqrt{x} + 5$?
  3. Find the range of $k(x) = -\sqrt{x-2} + 4$.

By tackling these problems, you'll sharpen your skills in analyzing functions and identifying their ranges. Remember to consider the properties of each component, such as the square root, constants, and any transformations applied to the function.

Final Thoughts

Determining the range of a function is a fundamental skill in mathematics, vital for understanding how functions behave and the possible outputs they can produce. By dissecting the function into its constituent parts, analyzing their properties, and considering how they interact, you can effectively determine the range. In the specific case of $g(x) = \sqrt{x-1} + 2$, we have demonstrated that the range is $y \geq 2$. This result stems from the non-negative nature of the square root function combined with the vertical shift induced by the constant term. Armed with these insights, you are well-prepared to tackle a wide array of problems involving function ranges. Understanding function range is crucial for mathematical analysis and problem-solving.

In mathematics, the concept of a function's range is crucial for understanding its behavior and output. The range defines the set of all possible values a function can produce. This article provides a detailed, step-by-step guide on how to determine the range of the function $g(x) = \sqrt{x-1} + 2$. We will dissect the function, analyze its components, and apply logical reasoning to arrive at the correct solution. By following this guide, you'll gain a solid understanding of range determination techniques, a fundamental skill in mathematical analysis.

Breaking Down the Function g(x) = √(x-1) + 2

To accurately determine the range, it's essential to break down the function $g(x) = \sqrt{x-1} + 2$ into its individual components. This function comprises a square root term and a constant term, each with its unique properties that influence the overall output. By understanding how these components interact, we can effectively determine the function's range. Let's analyze each component in detail.

The Square Root Term √(x-1) A Key Component

The square root function, denoted by $\sqrt{ }$, is the first critical element to consider. A fundamental property of the square root function is that it produces only non-negative values. This means the output of $\sqrt{x-1}$ will always be greater than or equal to zero. This stems from the definition of the square root: it yields the non-negative value that, when multiplied by itself, equals the number under the root. This non-negativity is a cornerstone in range determination for this function.

Moreover, we must consider the expression within the square root, which is x-1. For the square root to be defined within the realm of real numbers, the expression x-1 must be greater than or equal to zero. We cannot take the square root of a negative number and obtain a real number result. This leads to the inequality:

x−1≥0x - 1 \geq 0

Solving for x, we find:

x≥1x \geq 1

This inequality defines the domain of the function $g(x)$, which is all x values greater than or equal to 1. This restriction on input values plays a crucial role in shaping the possible output values and, consequently, the range of the function.

Given the non-negativity of the square root, the smallest possible value for $\sqrt{x-1}$ is 0. This occurs when x = 1, as $\sqrt{1-1} = \sqrt{0} = 0$. As x increases beyond 1, the value of $\sqrt{x-1}$ also increases. Importantly, the square root term will never be negative, establishing a lower boundary for the range.

Understanding the Constant Term +2

The constant term, +2, is the second key component, and its role is to vertically shift the function's output. This term adds 2 to the value produced by the square root component. Graphically, this translates to shifting the entire function's graph upwards by 2 units. Recognizing this vertical shift is crucial for accurately determining the function's range. The constant term directly influences the minimum possible output value, effectively setting a lower limit for the range.

Piecing It Together Determining the Range of g(x)

Now that we've analyzed the individual components, we can synthesize our understanding to determine the range of $g(x) = \sqrt{x-1} + 2$. We know that the square root term, $\sqrt{x-1}$, is always greater than or equal to 0. This implies that the smallest possible value for $\sqrt{x-1}$ is 0.

When $\sqrt{x-1} = 0$, the function evaluates to:

g(x)=0+2=2g(x) = 0 + 2 = 2

This crucial calculation reveals that the smallest possible output value (y-value) of the function is 2. Since the square root term can only increase as x increases (for x ≥ 1), the output of the function will always be greater than or equal to 2. As x grows larger, the value of $\sqrt{x-1}$ also increases, leading to larger output values for $g(x)$. There's no upper limit to how large the output can become, signifying that the range extends infinitely upwards. Therefore, the range of the function encompasses all y-values greater than or equal to 2.

Expressing the Range Using Different Notations

The range of the function $g(x) = \sqrt{x-1} + 2$ can be expressed using various notations, each conveying the same information in a slightly different way:

  • Inequality Notation: $y \geq 2$
  • Interval Notation: $[2, \infty)$

Both notations effectively communicate that the range of the function includes all real numbers y that are greater than or equal to 2. The inequality notation directly states this condition, while the interval notation uses a bracket to indicate that 2 is included in the range and the infinity symbol to represent the unbounded nature of the upper limit.

Arriving at the Correct Answer

Based on our thorough analysis, the correct answer to the question "What is the range of function $g(x) = \sqrt{x-1} + 2$?" is:

C. y ≥ 2

This conclusion aligns perfectly with our meticulous examination of the function's components and their influence on the output values. The non-negative square root term establishes a lower bound, and the constant term vertically shifts the entire function, reinforcing the range.

Visualizing the Range Graphically

To enhance comprehension, visualizing the function's graph can be immensely helpful. The graph of $g(x) = \sqrt{x-1} + 2$ originates at the point (1, 2) and extends upwards and to the right. The y-values on the graph represent the function's output, and you'll observe that they all lie above or on the line y = 2. This visual representation serves as a concrete confirmation of the range we derived analytically.

Key Concepts and Takeaways to Remember

  • The range of a function represents the set of all possible output values (y-values).
  • The square root function ($\sqrt{ }$) produces only non-negative values.
  • The expression inside the square root must be greater than or equal to zero for the function to be defined in the real number system.
  • Constant terms in a function cause vertical shifts in the graph, affecting the range.
  • The range of $g(x) = \sqrt{x-1} + 2$ is $y \geq 2$, signifying that the output values are always greater than or equal to 2.

Practice Exercises for Solidifying Your Understanding

To reinforce your understanding of determining the range of functions, try tackling these practice exercises:

  1. Determine the range of the function $f(x) = \sqrt{x+3} - 1$.
  2. What is the range of $h(x) = 2\sqrt{x} + 5$?
  3. Find the range of $k(x) = -\sqrt{x-2} + 4$.

By engaging with these problems, you'll hone your skills in analyzing functions and identifying their ranges. Remember to consider the properties of each component, including the square root, constants, and any transformations applied to the function.

In Conclusion A Mastering Function Range

Determining the range of a function is a fundamental skill in mathematics, essential for understanding function behavior and potential outputs. By dissecting the function into its constituent parts, analyzing their properties, and considering their interactions, you can effectively determine the range. In the specific case of $g(x) = \sqrt{x-1} + 2$, we've clearly demonstrated that the range is $y \geq 2$. This result stems from the non-negative characteristic of the square root function coupled with the vertical shift induced by the constant term. Equipped with these insights, you are well-prepared to tackle a broad spectrum of problems involving function ranges. Mastering function range determination is a valuable asset in mathematical analysis and problem-solving.