Determining The Range Of F(x) = -|x| - 3 A Step-by-Step Guide
Determining the range of a function is a fundamental concept in mathematics, particularly in algebra and calculus. The range represents the set of all possible output values (y-values) that a function can produce. This guide provides a comprehensive explanation of how to determine the range of the function f(x) = -|x| - 3. We will explore the properties of absolute value functions, transformations, and graphical analysis to understand the behavior of this function and accurately identify its range.
Understanding Absolute Value Functions
Absolute value functions play a crucial role in determining the range of f(x) = -|x| - 3. The absolute value of a number, denoted as |x|, is its distance from zero on the number line. Mathematically, the absolute value function is defined as:
|x| = x, if x ≥ 0 |x| = -x, if x < 0
The absolute value function always returns a non-negative value. This property is critical when analyzing the function f(x) = -|x| - 3. To fully grasp the impact of the absolute value in our function, let's delve deeper into its characteristics and how it affects the output.
The basic absolute value function, y = |x|, forms a V-shaped graph with its vertex at the origin (0, 0). The graph is symmetric about the y-axis because |x| = |-x| for all x. Understanding this symmetry is vital for visualizing how transformations affect the function's range. When we introduce a negative sign in front of the absolute value, as in -|x|, the graph is reflected across the x-axis. This reflection means that instead of opening upwards, the V-shape opens downwards, significantly altering the possible output values.
Consider the function g(x) = |x|. The range of g(x) is [0, ∞) because the absolute value ensures that the output is always non-negative. However, when we look at -|x|, the outputs become non-positive, meaning the range shifts to (-∞, 0]. This change is a direct consequence of the reflection. Recognizing this transformation is essential for determining the range of more complex functions involving absolute values.
In the context of f(x) = -|x| - 3, we have two transformations to consider: the reflection due to the negative sign and the vertical shift due to the subtraction of 3. The reflection, as discussed, flips the graph downwards. The vertical shift then moves the entire graph down by 3 units. These transformations collectively dictate the final range of the function. Visualizing these transformations step-by-step helps in accurately predicting the function's behavior and its possible output values.
Understanding the fundamental properties of the absolute value function, such as its non-negativity and symmetry, is essential. By recognizing how transformations like reflections and vertical shifts alter the function, we can more effectively determine its range. In the subsequent sections, we will apply these principles to analyze f(x) = -|x| - 3 in detail and find its range.
Transformations of f(x) = -|x| - 3
Analyzing the transformations applied to the basic absolute value function is essential to determining the range of f(x) = -|x| - 3. We can break down the function into two primary transformations: a reflection over the x-axis and a vertical shift. These transformations significantly impact the function's output values and, consequently, its range.
First, let's consider the reflection. The negative sign in front of the absolute value, -|x|, reflects the graph of y = |x| over the x-axis. The graph of y = |x| is a V-shape that opens upwards, with its vertex at the origin (0, 0). The range of y = |x| is [0, ∞), meaning all output values are non-negative. However, when we apply the reflection, the graph flips downwards. This means that the output values become non-positive, and the range of y = -|x| is (-∞, 0]. The reflection transformation is a critical step in understanding how the function's outputs change from non-negative to non-positive.
The second transformation is the vertical shift. The term -3 in f(x) = -|x| - 3 shifts the graph of y = -|x| downward by 3 units. A vertical shift alters the position of the graph along the y-axis, affecting the function's minimum or maximum value and, consequently, its range. Starting with the range of y = -|x|, which is (-∞, 0], subtracting 3 from every output value shifts the entire range downward by 3 units. This shift means that the new range will extend from negative infinity up to -3, inclusive.
To visualize these transformations, imagine starting with the basic absolute value function, y = |x|. Reflecting it over the x-axis gives us y = -|x|, which now has a maximum value of 0 at x = 0. Shifting this reflected graph down by 3 units results in the function f(x) = -|x| - 3. The vertex of the V-shape, which was at (0, 0) for y = |x|, moves to (0, -3). The graph now opens downwards, and its highest point is at y = -3. This visual representation makes it clear that the maximum value of f(x) is -3, and all other values are less than or equal to -3.
By understanding these transformations, we can deduce that the range of f(x) = -|x| - 3 is (-∞, -3]. The reflection over the x-axis ensures that the output values are non-positive, and the vertical shift by -3 units lowers the entire range by 3 units. This step-by-step analysis of transformations is a powerful method for determining the range of functions, especially those involving absolute values.
In summary, analyzing transformations helps us understand how each component of the function affects its graph and, ultimately, its range. The reflection over the x-axis and the vertical shift are critical in determining that the range of f(x) = -|x| - 3 is (-∞, -3]. In the next section, we will use graphical analysis to confirm this range and further illustrate the behavior of the function.
Graphical Analysis of f(x) = -|x| - 3
Graphical analysis is a powerful tool for understanding the behavior of a function and determining its range. By plotting the function f(x) = -|x| - 3, we can visually confirm the transformations discussed earlier and accurately identify the range. A graph provides a clear representation of all possible output values, making it easier to understand the function's overall behavior.
To plot the function f(x) = -|x| - 3, we can start by considering key points and the transformations involved. The basic absolute value function, y = |x|, has a V-shape with its vertex at the origin (0, 0). The function is symmetric about the y-axis, with its arms extending upwards. When we apply the transformations, the shape changes significantly.
The first transformation, the reflection over the x-axis due to the negative sign, flips the V-shape downwards. The function y = -|x| now opens downwards, with its vertex still at (0, 0). This reflection changes the possible output values from non-negative to non-positive. The range of y = -|x| is (-∞, 0], which means that the function can take any value less than or equal to 0.
The second transformation, the vertical shift of -3 units, moves the entire graph down. The function f(x) = -|x| - 3 now has its vertex at (0, -3). The V-shape opens downwards, and the highest point on the graph is at y = -3. All other points on the graph are below this maximum value. This vertical shift is crucial in defining the upper bound of the range.
Plotting a few key points can help visualize the graph accurately. For x = 0, f(0) = -|0| - 3 = -3. For x = 1, f(1) = -|1| - 3 = -4. For x = -1, f(-1) = -|-1| - 3 = -4. These points confirm the V-shape opening downwards with the vertex at (0, -3). As x moves away from 0 in either direction, the function values decrease, indicating that the graph extends downwards indefinitely.
By observing the graph, it becomes evident that the function f(x) = -|x| - 3 never takes values greater than -3. The maximum value of the function is -3, which occurs at x = 0. The graph extends downwards without bound, indicating that the function can take any value less than or equal to -3. Therefore, the range of f(x) = -|x| - 3 is (-∞, -3].
Graphical analysis provides a clear and intuitive way to understand the range of a function. By plotting the function and observing its behavior, we can visually confirm the effects of transformations and accurately identify the set of all possible output values. In the case of f(x) = -|x| - 3, the graph clearly shows that the range is (-∞, -3], confirming our analysis of the transformations.
In conclusion, graphical analysis is a valuable tool for determining the range of functions. By plotting the function and observing its behavior, we can visually confirm the effects of transformations and accurately identify the set of all possible output values. The graph of f(x) = -|x| - 3 clearly illustrates that the range is (-∞, -3], supporting our earlier analysis.
Determining the Range Algebraically
Determining the range algebraically provides a rigorous method to confirm the range of f(x) = -|x| - 3. By understanding the properties of the absolute value and applying algebraic manipulations, we can deduce the set of all possible output values without relying solely on graphical representations. This approach is particularly useful for functions where graphical analysis might be complex or less precise.
The first step in determining the range algebraically is to recognize the inherent properties of the absolute value function. As previously discussed, |x| is always non-negative, meaning |x| ≥ 0 for all real numbers x. This fundamental property is the cornerstone of our algebraic analysis. Since |x| is always greater than or equal to zero, multiplying it by -1 will reverse the inequality. Therefore, -|x| ≤ 0.
Now, consider the function f(x) = -|x| - 3. We know that -|x| is always less than or equal to 0. To incorporate the -3 term, we subtract 3 from both sides of the inequality: -|x| - 3 ≤ 0 - 3, which simplifies to -|x| - 3 ≤ -3. This inequality tells us that the output values of f(x) can never be greater than -3. This establishes an upper bound for the range.
To fully define the range, we need to consider if f(x) can take on all values less than or equal to -3. Let's analyze this further. Since |x| can take any non-negative value, -|x| can take any non-positive value. This means that -|x| can be 0, -1, -2, and so on. When we subtract 3 from these values, we get -3, -4, -5, and so on. This indicates that f(x) can indeed take any value less than or equal to -3.
To express this mathematically, let y be any real number such that y ≤ -3. We want to show that there exists an x such that f(x) = y. Consider the equation -|x| - 3 = y. Adding 3 to both sides gives -|x| = y + 3. Since y ≤ -3, y + 3 ≤ 0. Multiplying both sides by -1 yields |x| = -(y + 3). Because y + 3 is non-positive, -(y + 3) is non-negative, which means that |x| can equal this value.
We can find a solution for x by setting x = ±[-(y + 3)]. For example, if y = -5, then |x| = -(-5 + 3) = -(-2) = 2, so x can be 2 or -2. This confirms that for any y ≤ -3, there exists an x such that f(x) = y. Therefore, the range of f(x) is all real numbers less than or equal to -3.
Thus, algebraically, we have demonstrated that the range of f(x) = -|x| - 3 is (-∞, -3]. This aligns with our findings from both the transformation analysis and the graphical analysis. The algebraic method provides a formal and rigorous approach to determining the range, reinforcing our understanding of the function's behavior.
In summary, the algebraic method involves understanding the properties of absolute value, manipulating inequalities, and demonstrating that for any y in the proposed range, there exists an x such that f(x) = y. This method confirms that the range of f(x) = -|x| - 3 is (-∞, -3], providing a comprehensive understanding of the function's output values.
Conclusion
In conclusion, determining the range of the function f(x) = -|x| - 3 involves a comprehensive understanding of absolute value functions, transformations, graphical analysis, and algebraic methods. Throughout this guide, we have explored these concepts in detail to provide a clear and thorough explanation of how to find the range.
First, we discussed the properties of absolute value functions, emphasizing that |x| always returns a non-negative value. This understanding is crucial because it forms the foundation for analyzing how transformations affect the function's output values. We highlighted that the basic absolute value function, y = |x|, has a V-shape with its vertex at the origin and a range of [0, ∞).
Next, we examined the transformations applied to f(x) = -|x| - 3. We identified two primary transformations: a reflection over the x-axis due to the negative sign and a vertical shift of -3 units. The reflection flips the V-shape downwards, changing the range to (-∞, 0]. The vertical shift then moves the graph down by 3 units, further altering the range. By understanding these transformations, we deduced that the range of f(x) = -|x| - 3 is (-∞, -3].
Graphical analysis provided a visual confirmation of our findings. By plotting the function, we observed that the graph opens downwards with its vertex at (0, -3). The graph extends downwards without bound, indicating that the function can take any value less than or equal to -3. This graphical representation clearly supported our conclusion that the range is (-∞, -3].
Finally, we employed an algebraic method to rigorously confirm the range. By using the property that |x| ≥ 0, we showed that -|x| ≤ 0. Then, by subtracting 3 from both sides, we established that -|x| - 3 ≤ -3. This inequality demonstrates that the output values of f(x) can never be greater than -3. We also showed that for any y ≤ -3, there exists an x such that f(x) = y, thus confirming that the range is indeed (-∞, -3].
By combining these approaches, we have provided a comprehensive guide to determining the range of f(x) = -|x| - 3. Each method—understanding absolute value properties, analyzing transformations, using graphical analysis, and applying algebraic techniques—contributes to a thorough understanding of the function's behavior and its possible output values. This multi-faceted approach ensures accuracy and reinforces the underlying mathematical principles.
Understanding the range of a function is a fundamental skill in mathematics. By mastering these techniques, students and practitioners can effectively analyze and interpret various functions, leading to a deeper understanding of mathematical concepts and their applications in real-world scenarios. The function f(x) = -|x| - 3 serves as an excellent example for illustrating these methods and solidifying the understanding of range determination.
In summary, the range of f(x) = -|x| - 3 is (-∞, -3], a result we have confidently confirmed through various analytical methods. This guide serves as a valuable resource for anyone seeking to enhance their understanding of function ranges and mathematical analysis.
