Finding The Range Of F(x) = -4|x + 1| - 5 A Comprehensive Guide
What is the range of the function f(x)=-4|x+1|-5? Explain how to find the range of the function.
Determining the range of a function is a fundamental concept in mathematics, particularly in the study of functions and their behavior. The range represents the set of all possible output values (y-values) that a function can produce. In this article, we will delve into finding the range of the function f(x) = -4|x + 1| - 5. We will explore the properties of absolute value functions, how transformations affect them, and ultimately, how to identify the range of the given function. Before we dive into the specifics of this function, it’s crucial to grasp the basics of absolute value functions and their general form. The absolute value function, denoted as |x|, returns the non-negative magnitude of a real number, regardless of its sign. For instance, |3| = 3 and |-3| = 3. This characteristic gives the absolute value function its distinctive V-shape when graphed, with the vertex at the origin (0, 0). Understanding the transformations applied to this basic form is key to determining the range of more complex absolute value functions.
Absolute Value Functions and Transformations
To fully understand how to find the range of f(x) = -4|x + 1| - 5, it’s essential to break down the transformations applied to the basic absolute value function, |x|. These transformations include stretches, compressions, reflections, and translations, each of which alters the graph and consequently the range of the function. Let's consider the general form a|x - h| + k, where a, h, and k are constants that dictate these transformations. The constant a determines the vertical stretch or compression and any reflection across the x-axis. If |a| > 1, the graph is stretched vertically, making it narrower. If 0 < |a| < 1, the graph is compressed vertically, making it wider. When a is negative, the graph is reflected across the x-axis, inverting the V-shape. The constant h represents a horizontal translation. If h is positive, the graph shifts to the right by h units, and if h is negative, the graph shifts to the left by |h| units. This shift affects the position of the vertex of the absolute value function. Finally, the constant k represents a vertical translation. A positive k shifts the graph upward by k units, while a negative k shifts it downward by |k| units. The vertical translation directly affects the minimum or maximum value of the function, which is crucial in determining the range. By understanding these transformations, we can visualize how the graph of the absolute value function is manipulated and how these manipulations affect the set of possible output values. For instance, a reflection across the x-axis coupled with a vertical shift downward will result in a range that extends to negative infinity, bounded by a maximum value. This comprehensive understanding of transformations sets the stage for analyzing the specific function f(x) = -4|x + 1| - 5 and identifying its range.
Analyzing f(x) = -4|x + 1| - 5
Now, let's apply our understanding of absolute value functions and transformations to the specific function f(x) = -4|x + 1| - 5. This function is a transformed version of the basic absolute value function, and by carefully examining each component, we can determine its range. First, we can identify the value of a as -4. The negative sign indicates a reflection across the x-axis, which means the V-shape of the absolute value function will be inverted, opening downwards. The magnitude of a, which is 4, indicates a vertical stretch. This stretch makes the graph steeper compared to the basic absolute value function. Next, we look at the term (x + 1) inside the absolute value. This corresponds to a horizontal translation. Since we have (x + 1), which can be written as (x - (-1)), this means the graph is shifted 1 unit to the left. The vertex of the basic absolute value function at (0, 0) is now shifted to (-1, 0) before considering the vertical shift. Finally, we have the constant term -5, which represents a vertical translation. This term shifts the entire graph 5 units downward. Consequently, the vertex of the transformed function moves from (-1, 0) to (-1, -5). Considering all these transformations, we can visualize the graph of f(x) as an inverted V-shape, stretched vertically, with its vertex at the point (-1, -5). The reflection across the x-axis means the function will have a maximum value, and the vertical shift downward means this maximum value will be -5. Understanding these transformations allows us to conclude that the function's output values will be all real numbers less than or equal to -5. This sets the stage for formally defining the range of the function.
Determining the Range
Based on our analysis of the transformations applied to the absolute value function f(x) = -4|x + 1| - 5, we can now precisely determine its range. We've established that the function is an inverted V-shape due to the negative coefficient of the absolute value term, and it has been shifted vertically downward by 5 units. This means the vertex of the function, which represents the maximum point, is at (-1, -5). Since the function opens downwards, the y-values will extend from -5 downwards to negative infinity. To express this mathematically, we can say that the range of f(x) includes all y-values such that y ≤ -5. In interval notation, this is represented as (-∞, -5]. The parenthesis on the negative infinity side indicates that negative infinity is not a specific number but a concept of unboundedness, while the square bracket on the -5 side indicates that -5 is included in the range. We can verify this by considering the properties of the absolute value function. The absolute value term, |x + 1|, is always non-negative, meaning it is greater than or equal to 0 for all values of x. When we multiply this by -4, we get a value that is always non-positive, meaning it is less than or equal to 0. Therefore, -4|x + 1| ≤ 0. Subtracting 5 from this inequality, we get -4|x + 1| - 5 ≤ -5. This confirms that the maximum value of the function is -5, and all other values are less than or equal to -5. Thus, the range of the function f(x) = -4|x + 1| - 5 is indeed (-∞, -5]. This process of analyzing transformations and understanding the properties of absolute value functions provides a robust method for determining the range of such functions.
Conclusion
In conclusion, finding the range of the function f(x) = -4|x + 1| - 5 involves a thorough understanding of absolute value functions and the transformations applied to them. By recognizing the reflection across the x-axis, the vertical stretch, the horizontal translation, and the vertical translation, we were able to determine that the function opens downwards with a maximum value at -5. This led us to the conclusion that the range of the function is all real numbers less than or equal to -5, which is expressed in interval notation as (-∞, -5]. The key takeaways from this analysis include the importance of understanding how transformations affect the graph of a function and how the properties of the absolute value function constrain its output values. The range of a function is a critical aspect of its behavior, and mastering the techniques to find it is essential for further studies in mathematics, including calculus and analysis. The ability to analyze functions and determine their ranges is not only a valuable skill in academics but also has practical applications in various fields, such as engineering, economics, and computer science. Understanding the range helps in predicting the possible outcomes of a model or system represented by the function. Therefore, a strong grasp of these concepts is crucial for both theoretical and applied mathematics. In essence, this exploration of the range of f(x) = -4|x + 1| - 5 serves as a valuable exercise in understanding the behavior of functions and their transformations, reinforcing the fundamental principles of mathematical analysis.