Find The Range Of The Function G(x) = |x-6| - 8

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The function presented, g(x) = |x-6| - 8, involves the absolute value, which significantly influences its range. To accurately determine the range, we need to understand how the absolute value function transforms the input and affects the output. This article delves into a comprehensive analysis of this function, exploring the properties of absolute values and their impact on graphical representation, culminating in a clear understanding of how to identify the range. Understanding the range of a function is crucial in mathematics, as it defines the set of all possible output values. This concept is fundamental not only in algebra and calculus but also in real-world applications where functions are used to model various phenomena. Mastering the ability to determine the range of functions, particularly those involving absolute values, empowers us to solve complex problems and interpret mathematical models effectively.

Dissecting the Absolute Value Function

At the heart of our function lies the absolute value expression, |x-6|. The absolute value of a number is its distance from zero, meaning it is always non-negative. This key characteristic shapes the behavior of the entire function. Let's break down how this works in the context of our function.

The Role of |x-6|

Consider the expression inside the absolute value, x-6. This represents a horizontal shift of the basic absolute value function |x| by 6 units to the right. This shift changes the vertex (the lowest point) of the graph. The absolute value then takes the result of this shift and makes it non-negative. For example:

  • If x = 6, then |x-6| = |6-6| = |0| = 0
  • If x = 10, then |x-6| = |10-6| = |4| = 4
  • If x = 2, then |x-6| = |2-6| = |-4| = 4

Notice that the output of |x-6| is always greater than or equal to zero. This is a fundamental property of the absolute value function. This non-negativity is the starting point for determining the range of the overall function. The absolute value function |x-6| acts as a foundational element, ensuring that the initial part of our composite function will never produce negative outputs. This constraint is crucial because it directly influences the final set of possible y-values, shaping the lower bound of the range. Understanding this behavior allows us to predict how subsequent transformations, such as the subtraction of 8, will further affect the function's range. The non-negative nature of the absolute value is the cornerstone of determining the final range, making it essential to focus on this property during analysis.

The Impact of the Constant Term -8

The next crucial component is the “-8” term. This term represents a vertical shift of the entire graph downwards by 8 units. This vertical shift directly affects the range of the function. The constant term is the key to finding the lower boundary of the function's range. Subtracting 8 from the absolute value expression means that every output of |x-6| will be reduced by 8. Since the minimum value of |x-6| is 0, the minimum value of g(x) = |x-6| - 8 will be 0 - 8 = -8. This establishes that no output of the function can be less than -8. The vertical shift caused by the constant term reshapes the position of the graph on the coordinate plane, directly influencing the set of y-values that the function can achieve. Understanding this vertical translation is essential for accurately determining the range because it defines the lower limit of the possible output values.

Graphing the Function for Visual Insight

Visualizing the graph of g(x) = |x-6| - 8 provides an intuitive understanding of its range. The graph of y = |x| is a V-shaped graph with its vertex (the point where the two lines meet) at the origin (0,0). The transformations we discussed earlier affect this basic shape.

Transformations in Action

  1. Horizontal Shift: The |x-6| part shifts the V-shaped graph 6 units to the right. The vertex is now at the point (6,0).
  2. Vertical Shift: The “-8” part shifts the entire graph 8 units downwards. This moves the vertex to the point (6,-8).

The resulting graph is a V-shaped graph with its vertex at (6,-8). The graph opens upwards, extending infinitely in the positive y-direction. The visualization of the transformations helps in understanding how each component of the function contributes to its final shape and position. The horizontal shift repositions the function along the x-axis, while the vertical shift alters its height on the y-axis. The final vertex position, in this case, (6,-8), becomes the lowest point of the graph and the key determinant of the range. Seeing the graph extend upwards indefinitely further clarifies that there is no upper bound to the output values, solidifying the understanding of the function's range.

Interpreting the Graph for the Range

The graph vividly illustrates that the lowest y-value the function attains is -8 (at the vertex). Since the graph extends upwards indefinitely, there is no upper bound on the y-values. Therefore, the range of the function includes -8 and all values greater than -8. Observing the graph, it's evident that every y-value above -8 is covered by the function's trace, and no y-value falls below this limit. This direct visual interpretation reinforces the mathematical analysis and confirms the range of the function. The graph acts as a powerful tool, turning an abstract concept into a concrete visual representation, making the determination of the range more intuitive and less prone to errors.

Determining the Range Algebraically

While graphing provides a visual aid, we can also determine the range algebraically by considering the properties of the function.

Minimum Value of the Absolute Value

As we established, the absolute value |x-6| is always greater than or equal to zero. The minimum value of |x-6| is 0, which occurs when x = 6. This is a foundational aspect of understanding absolute value functions and is critical for algebraic manipulation. Recognizing that the absolute value expression is bounded from below by zero allows us to establish a starting point for determining the overall function's range. We can then analyze how subsequent operations, such as subtracting 8, affect this lower bound, ultimately leading to the identification of the minimum y-value and, thus, the range of the function.

Applying the Vertical Shift

Since the minimum value of |x-6| is 0, the minimum value of g(x) = |x-6| - 8 is 0 - 8 = -8. This confirms that -8 is the lowest possible output value. The vertical shift directly dictates the lower limit of the range. Subtracting 8 shifts the entire range downward, meaning that the minimum y-value of the transformed function will be 8 units less than the minimum y-value of the original absolute value function. This straightforward algebraic manipulation allows us to pinpoint the lower boundary of the range with precision.

Defining the Range

Since |x-6| can take any non-negative value, g(x) = |x-6| - 8 can take any value greater than or equal to -8. Therefore, the range of the function is {y | y ≥ -8}. The algebraic method provides a robust and direct way to express the range, ensuring clarity and accuracy in mathematical communication. It complements the visual interpretation from the graph, solidifying our understanding and reinforcing the correctness of the determined range.

Conclusion

By carefully analyzing the function g(x) = |x-6| - 8, both graphically and algebraically, we have determined that its range is {y | y ≥ -8}. This means that the function's output values include -8 and all real numbers greater than -8. This comprehensive exploration of the function, involving both visualization and algebraic manipulation, provides a solid foundation for understanding the properties of absolute value functions and their ranges. The process reinforces the importance of breaking down complex functions into simpler components, analyzing the transformations, and systematically determining the set of possible output values. This approach is invaluable for solving a wide range of mathematical problems and fostering a deeper understanding of functional behavior.

Therefore, the correct answer is B. {y | y ≥ -8}.

Keywords: range of a function, absolute value function, vertical shift, horizontal shift, graph of a function, minimum value, g(x) = |x-6| - 8, algebraic determination, visual interpretation.