Functions With Matching Ranges Finding G(x) For F(x)=-2√(x-3)+8

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Determining the range of a function is a fundamental concept in mathematics, crucial for understanding the function's behavior and its possible output values. When comparing functions, identifying those with the same range is a common problem, often requiring a solid grasp of transformations and how they affect a function's output. In this article, we will delve into the process of finding a function with the same range as the given function, f(x) = -2√(x-3) + 8. We'll explore the key concepts, discuss different transformation types, and provide a step-by-step analysis to solve this problem effectively.

Decoding the Range of f(x) = -2√(x-3) + 8

To begin, let's analyze the given function, f(x) = -2√(x-3) + 8, and determine its range. The range of a function represents the set of all possible output values (y-values) that the function can produce. For this function, we need to consider the transformations applied to the basic square root function, √x. Understanding these transformations is critical for accurately determining the range.

Breaking Down the Transformations

The function f(x) = -2√(x-3) + 8 involves several transformations applied to the basic square root function, √x:

  1. Horizontal Shift: The term (x-3) inside the square root indicates a horizontal shift. Specifically, the graph of √x is shifted 3 units to the right. This shift affects the domain of the function but not the range directly.
  2. Vertical Stretch and Reflection: The coefficient -2 in front of the square root signifies two transformations. The absolute value, 2, represents a vertical stretch by a factor of 2, making the graph vertically elongated. The negative sign indicates a reflection across the x-axis, flipping the graph upside down. This reflection is crucial as it inverts the range of the basic square root function.
  3. Vertical Shift: The constant term +8 represents a vertical shift. In this case, the graph is shifted 8 units upward. This shift directly affects the range by raising the entire graph.

Determining the Range Step-by-Step

Let's trace the transformations to determine the range:

  1. The basic square root function, √x, has a range of [0, ∞), meaning it outputs all non-negative real numbers.
  2. The vertical stretch by a factor of 2 does not change the range, so it remains [0, ∞).
  3. The reflection across the x-axis inverts the range, changing it to (-∞, 0]. This means the function now outputs all non-positive real numbers.
  4. Finally, the vertical shift of 8 units upward adds 8 to every value in the range, resulting in the range (-∞, 8]. This indicates that the function's output values are all real numbers less than or equal to 8.

Therefore, the range of f(x) = -2√(x-3) + 8 is (-∞, 8]. This means the function's output values include all real numbers less than or equal to 8.

Analyzing the Answer Choices

Now that we know the range of f(x) = -2√(x-3) + 8 is (-∞, 8], let's examine the given answer choices to determine which function has the same range. We'll analyze each option by identifying the transformations applied to the basic square root function and how they affect the range.

A. g(x) = √(x-3) - 8

This function, g(x) = √(x-3) - 8, involves a horizontal shift and a vertical shift.

  • The (x-3) term shifts the graph 3 units to the right, affecting the domain but not the range.
  • The -8 term shifts the graph 8 units downward.

The basic square root function, √x, has a range of [0, ∞). The vertical shift of -8 units changes the range to [-8, ∞). This range is different from the range of f(x), which is (-∞, 8]. Therefore, option A is not the correct answer.

B. g(x) = √(x-3) + 8

This function, g(x) = √(x-3) + 8, also involves a horizontal shift and a vertical shift.

  • The (x-3) term shifts the graph 3 units to the right, affecting the domain but not the range.
  • The +8 term shifts the graph 8 units upward.

Starting with the basic square root function's range of [0, ∞), the vertical shift of +8 units changes the range to [8, ∞). This range is different from the range of f(x), which is (-∞, 8]. Therefore, option B is not the correct answer.

C. g(x) = -√(x+3) + 8

This function, g(x) = -√(x+3) + 8, involves a horizontal shift, a reflection across the x-axis, and a vertical shift.

  • The (x+3) term shifts the graph 3 units to the left, affecting the domain but not the range.
  • The negative sign in front of the square root reflects the graph across the x-axis, inverting the range.
  • The +8 term shifts the graph 8 units upward.

Starting with the basic square root function's range of [0, ∞), the reflection across the x-axis changes the range to (-∞, 0]. The vertical shift of +8 units then changes the range to (-∞, 8]. This range is the same as the range of f(x). Therefore, option C is the correct answer.

D. g(x) = -√(x-3) - 8

This function, g(x) = -√(x-3) - 8, involves a horizontal shift, a reflection across the x-axis, and a vertical shift.

  • The (x-3) term shifts the graph 3 units to the right, affecting the domain but not the range.
  • The negative sign in front of the square root reflects the graph across the x-axis, inverting the range.
  • The -8 term shifts the graph 8 units downward.

Starting with the basic square root function's range of [0, ∞), the reflection across the x-axis changes the range to (-∞, 0]. The vertical shift of -8 units then changes the range to (-∞, -8]. This range is different from the range of f(x), which is (-∞, 8]. Therefore, option D is not the correct answer.

Conclusion

By analyzing the transformations applied to the basic square root function in each option, we determined that only g(x) = -√(x+3) + 8 has the same range as f(x) = -2√(x-3) + 8, which is (-∞, 8]. Therefore, the correct answer is option C.

Understanding function transformations is key to identifying functions with identical ranges. By breaking down each transformation and its effect on the range, we can systematically compare functions and arrive at the correct answer. This process highlights the importance of a strong foundation in function analysis and graphical transformations.

To summarize, determining the range of a function and comparing it with other functions involves:

  • Identifying the transformations applied to the basic function.
  • Understanding how each transformation affects the range.
  • Tracing the changes in the range step-by-step.
  • Comparing the resulting range with the given function's range.

This approach enables us to effectively solve problems involving function ranges and transformations, reinforcing our understanding of function behavior and properties.

Additional Tips for Mastering Function Ranges

To further enhance your understanding and skills in determining function ranges, consider these additional tips:

  1. Visualize the Graphs: Sketching the graphs of the functions can provide a visual representation of the range. Use graphing tools or software to plot the functions and observe their output values. This visual aid can make it easier to identify the range and compare different functions.
  2. Identify Key Points: Pay attention to key points on the graph, such as the vertex (for quadratic functions), the starting point (for square root functions), and asymptotes (for rational functions). These points often define the boundaries of the range.
  3. Consider the Domain: While the range is the set of output values, the domain (the set of input values) can sometimes influence the range. For example, restrictions on the domain may limit the possible output values. Always consider the domain when determining the range.
  4. Practice with Various Functions: Work through a variety of examples involving different types of functions, such as linear, quadratic, exponential, logarithmic, trigonometric, and piecewise functions. This practice will help you develop a broader understanding of range determination techniques.
  5. Use Interval Notation: Express the range using interval notation. This notation provides a concise way to represent the set of all possible output values. Remember to use parentheses for open intervals (values not included) and brackets for closed intervals (values included).

By incorporating these tips into your study routine, you can strengthen your understanding of function ranges and improve your problem-solving skills in this area.

Practice Problems

To solidify your understanding of function ranges, try solving these practice problems:

  1. Determine the range of the function h(x) = 3(x-2)^2 + 5.
  2. Which of the following functions has the same range as k(x) = -|x+1| + 4?
    • A. m(x) = |x-1| - 4
    • B. m(x) = -|x+1| - 4
    • C. m(x) = |x+1| + 4
    • D. m(x) = -|x-1| + 4
  3. Find the range of the piecewise function:
    • p(x) = { x + 2, if x < 0; x^2, if 0 ≤ x ≤ 2 }

These problems will challenge you to apply the concepts and techniques discussed in this article. By working through these exercises, you'll gain confidence in your ability to determine function ranges and solve related problems.

Final Thoughts

Understanding function ranges is essential for a comprehensive understanding of function behavior and their applications in mathematics and other fields. By mastering the techniques for determining the range of various functions and comparing them, you'll be well-equipped to tackle more advanced concepts in calculus, analysis, and beyond. Remember to practice regularly, visualize graphs, and consider the transformations applied to the basic functions. With consistent effort, you'll develop a strong foundation in function ranges and their significance.

By consistently practicing and applying these concepts, you'll develop a strong understanding of function ranges and their significance in mathematics.