Determining The Range Of Absolute Value Function F(x) = -|x| - 3

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Introduction

The absolute value function is a fundamental concept in mathematics, particularly in algebra and calculus. Understanding how absolute value transformations affect the range of a function is crucial for problem-solving. In this article, we will delve into the specifics of the absolute value function f(x) = -|x| - 3, and meticulously determine its range. This involves analyzing how the absolute value, negation, and vertical translation collectively shape the function's output values. This exploration will not only provide a clear answer to the given problem but also reinforce your understanding of function transformations and their impact on the range. We'll break down each component of the function, starting with the basic absolute value function and then applying transformations step-by-step to reach our final function. By the end of this discussion, you should have a firm grasp on how to approach similar problems involving absolute value functions and their ranges.

Core Concepts of Absolute Value Functions

Before we dive into the specifics of our function, f(x) = -|x| - 3, it's important to understand the core concepts surrounding absolute value functions. The absolute value of a number is its distance from zero on the number line, irrespective of direction. Mathematically, the absolute value of x, denoted as |x|, is defined as x if x is non-negative and -x if x is negative. This seemingly simple definition has profound implications for the behavior of functions incorporating absolute values. The most basic absolute value function is f(x) = |x|, which produces a V-shaped graph with its vertex at the origin (0, 0). The range of this function is all non-negative real numbers, or [0, ∞), since the output is always zero or a positive number. Understanding this foundational concept is critical because transformations applied to this basic function will alter its shape and position, consequently affecting its range. For instance, multiplying the absolute value by a negative number, as seen in our function, reflects the graph across the x-axis, changing the direction of the opening of the “V”. Similarly, adding or subtracting constants shifts the graph vertically, altering the minimum or maximum value of the function. With these core concepts in mind, we are well-equipped to analyze the more complex function at hand.

Step-by-Step Analysis of f(x) = -|x| - 3

To accurately determine the range of the function f(x) = -|x| - 3, we need to dissect it step by step, understanding the impact of each transformation. Let's start with the basic absolute value function, g(x) = |x|. As mentioned earlier, this function produces non-negative outputs, with a range of [0, ∞). The graph is a V-shape opening upwards, with its vertex at the origin. Now, consider the transformation introduced by the negative sign in our function, making it h(x) = -|x|. This negation reflects the graph of g(x) across the x-axis. Consequently, the V-shape now opens downwards, and the outputs are no longer non-negative but non-positive. The range of h(x) becomes (-∞, 0], indicating that the function's values are zero or negative. This reflection is a critical step in understanding the final function. Finally, we consider the constant term in f(x) = -|x| - 3. The “- 3” represents a vertical shift, specifically a downward translation of 3 units. This shift takes the entire graph of h(x) and moves it down by 3 units. The vertex, which was at (0, 0) for g(x) and remained at (0, 0) after the reflection to become the highest point for h(x), now shifts to (0, -3). This vertical translation directly affects the range of the function. The range of f(x) = -|x| - 3 is therefore all real numbers less than or equal to -3, denoted as (-∞, -3]. This step-by-step analysis provides a clear and logical progression to understanding how the range is derived.

Determining the Range of f(x) = -|x| - 3

Based on our step-by-step analysis, we can confidently determine the range of the function f(x) = -|x| - 3. As we established, the basic absolute value function, |x|, has a range of [0, ∞). The negation in -|x| reflects the graph across the x-axis, flipping the range to (-∞, 0]. This means the maximum value of -|x| is 0. Now, the subtraction of 3, as in -|x| - 3, shifts the entire function vertically downwards by 3 units. This vertical shift directly impacts the range by decreasing all output values by 3. Consequently, the maximum value of the function, which was 0 after the reflection, is now -3. Since the function extends infinitely downwards due to the nature of the reflected absolute value, the range encompasses all real numbers less than or equal to -3. Therefore, the range of f(x) = -|x| - 3 is (-∞, -3]. This conclusion aligns with our understanding of vertical translations and their effect on the range of functions. By carefully tracking the transformations, we can accurately predict how the output values are affected. The range represents all possible y-values the function can take, and in this case, those values are all numbers on the number line from negative infinity up to and including -3.

Conclusion

In conclusion, by systematically analyzing the transformations applied to the basic absolute value function, we have determined that the range of f(x) = -|x| - 3 is all real numbers less than or equal to -3. This detailed examination underscores the importance of understanding how each component of a function—the absolute value, negation, and vertical translation—contributes to its overall behavior. Recognizing the foundational range of the absolute value function and then tracking the changes induced by reflections and shifts is crucial for accurately determining the range of more complex functions. This approach is not only applicable to absolute value functions but also extends to other types of functions and transformations in mathematics. Mastering these concepts will significantly enhance your problem-solving skills and deepen your understanding of function behavior. The ability to break down functions into their constituent parts and analyze their individual effects is a key skill in mathematical analysis, making it easier to tackle more intricate problems in the future. Therefore, understanding the range of transformed functions, like the one we've explored, is an essential aspect of mathematical proficiency.

The correct answer is D. all real numbers less than or equal to -3.