Determining The Range Of F(x) = -4|x+1|-5

In mathematics, understanding the range of a function is crucial for grasping its behavior and properties. The range represents the set of all possible output values that a function can produce. In this article, we will delve into determining the range of the function f(x) = -4|x+1| - 5. This function involves an absolute value expression, which adds an interesting dimension to the analysis. We will break down the function step by step, analyze the impact of each component, and arrive at the correct range. By the end of this exploration, you will have a solid understanding of how to find the range of functions involving absolute values and transformations.

Understanding the Absolute Value Function

At the heart of our function lies the absolute value expression |x+1|. The absolute value of a number is its distance from zero, regardless of the sign. This means that |x| is always non-negative; it's either zero or a positive number. The |x+1| part shifts the standard absolute value function one unit to the left. This transformation doesn't affect the fundamental property of the absolute value, which is its non-negativity. So, |x+1| will also always be greater than or equal to zero. This is a crucial point because it forms the foundation for understanding the range of the entire function.

To illustrate, let's consider a few examples. If x = -1, then |x+1| = |-1+1| = |0| = 0. If x = 0, then |x+1| = |0+1| = |1| = 1. If x = -2, then |x+1| = |-2+1| = |-1| = 1. Notice that regardless of whether x is greater or less than -1, the result of the absolute value is always non-negative. This behavior is fundamental to how the rest of the function will behave. Understanding this non-negativity is essential for correctly determining the range.

The Impact of the Coefficient -4

Now, let's consider the next part of the function: -4|x+1|. We're multiplying the absolute value expression by -4. This has two key effects. First, the multiplication by 4 stretches the absolute value function vertically. This means that the output values will be four times as large as they were before. Second, and more importantly, the multiplication by a negative number (-4) reflects the function across the x-axis. This reflection is critical because it changes the direction in which the function opens. Instead of opening upwards, as the standard absolute value function does, it now opens downwards.

Since |x+1| is always greater than or equal to zero, multiplying it by -4 will always result in a value that is less than or equal to zero. In other words, -4|x+1| will always be non-positive. The maximum value of -4|x+1| occurs when |x+1| is zero, which happens when x = -1. In this case, -4|x+1| = -4(0) = 0. For any other value of x, |x+1| will be positive, and multiplying it by -4 will yield a negative number. This means that the graph of -4|x+1| will be a V-shaped graph that opens downwards, with its vertex at the point where x = -1 and the value of the function is 0. This understanding is vital for determining the upper bound of the range of the function.

The Constant Term -5: A Vertical Shift

The final piece of our function is the constant term, -5. This term represents a vertical shift of the entire function. Subtracting 5 from -4|x+1| will shift the graph downwards by 5 units. This shift directly affects the range of the function, as it lowers all the output values by 5. The maximum value of -4|x+1| was 0, so when we subtract 5, the new maximum value becomes 0 - 5 = -5. This means that the vertex of the V-shaped graph, which was at ( -1, 0 ) for the function -4|x+1|, is now at ( -1, -5 ) for the function f(x) = -4|x+1| - 5.

Since the graph opens downwards, the maximum value of the function is -5. As x moves away from -1 in either direction, the value of |x+1| increases, and -4|x+1| becomes more negative. Subtracting 5 from an increasingly negative number makes the function value even more negative. This means that the function values will continue to decrease without any lower bound. Therefore, the function can take on any value less than or equal to -5. This understanding of the vertical shift is crucial for establishing the range's lower bound.

Determining the Range of f(x) = -4|x+1| - 5

Now that we have analyzed each component of the function, we can confidently determine its range. We know that the absolute value |x+1| is always non-negative. Multiplying it by -4 makes it non-positive, with a maximum value of 0. Finally, subtracting 5 shifts the entire graph downwards by 5 units, resulting in a maximum value of -5. Since the graph opens downwards, the function values will extend indefinitely in the negative direction. Therefore, the range of the function f(x) = -4|x+1| - 5 includes all real numbers less than or equal to -5.

In interval notation, this range is represented as (-∞, -5]. The parenthesis on the left side indicates that negative infinity is not included in the range, as it is a conceptual limit rather than a specific value. The square bracket on the right side indicates that -5 is included in the range, as it is the maximum value the function can attain. This notation accurately captures the set of all possible output values of the function. Understanding how to express the range in interval notation is an important aspect of mathematical communication.

Conclusion

In conclusion, the range of the function f(x) = -4|x+1| - 5 is (-∞, -5]. We arrived at this answer by carefully analyzing the impact of each component of the function: the absolute value, the coefficient -4, and the constant term -5. The absolute value ensures non-negativity, the coefficient reflects and stretches the graph, and the constant term shifts the graph vertically. By understanding these transformations, we can accurately determine the range of a function. This process highlights the importance of breaking down complex functions into simpler parts to fully understand their behavior. Mastering these techniques is essential for success in mathematics and related fields.

Analyzing the Options for the Range of f(x) = -4|x+1| - 5

When tackling problems involving the range of a function, it's crucial to not only understand the underlying concepts but also to carefully evaluate the given options. For the function f(x) = -4|x+1| - 5, we've established that the range is all real numbers less than or equal to -5. This means we're looking for an option that accurately represents this interval. Let's examine the provided options and see how they stack up against our understanding of the function's behavior.

A. (-∞, -5]

This option represents the set of all real numbers less than or equal to -5. The parenthesis indicates that negative infinity is not included, and the square bracket indicates that -5 is included. This aligns perfectly with our analysis of the function, where we determined that the maximum value is -5 and the function extends indefinitely in the negative direction. This looks like a promising candidate, but we'll continue to analyze the other options to ensure we make the correct selection.

B. [-5, ∞)

This option represents the set of all real numbers greater than or equal to -5. The square bracket indicates that -5 is included, and the parenthesis indicates that positive infinity is not included. This range would imply that the function's values are bounded below by -5 and extend upwards. However, we know that our function opens downwards due to the negative coefficient in front of the absolute value. Therefore, this option is incorrect because it contradicts the behavior we've observed.

C. [-4, ∞)

This option represents the set of all real numbers greater than or equal to -4. Similar to option B, this range suggests that the function's values are bounded below. However, -4 is not a relevant value in the context of our function's range. The vertical shift of -5 plays a critical role in defining the range, and this option overlooks that. Thus, this option is incorrect because it doesn't align with our understanding of the function's vertical shift and direction.

D. (-∞, -4]

This option represents the set of all real numbers less than or equal to -4. While it correctly indicates that the range extends in the negative direction, it incorrectly places the upper bound at -4. We know that the maximum value of the function is -5, not -4. This option fails to account for the precise impact of the constant term -5 on the function's range. Therefore, this option is also incorrect.

Selecting the Correct Answer

After carefully analyzing each option, it's clear that A. (-∞, -5] is the only one that accurately represents the range of the function f(x) = -4|x+1| - 5. This option correctly captures the fact that the function's values are less than or equal to -5 and extend indefinitely in the negative direction. The other options either misrepresent the direction of the function's opening or misplace the upper bound of the range.

By systematically evaluating each option and comparing it to our understanding of the function's behavior, we can confidently select the correct answer. This process reinforces the importance of not only understanding the underlying concepts but also carefully applying them to the specific problem at hand. Choosing the right answer involves a thorough understanding and evaluation of all possibilities.

In conclusion, the correct answer for the range of the function f(x) = -4|x+1| - 5 is A. (-∞, -5]. This article has provided a comprehensive exploration of how to determine the range of such a function, emphasizing the importance of understanding the individual components and their combined effects. We dissected the absolute value, the coefficient, and the constant term, demonstrating how each contributes to the function's overall behavior and range. This step-by-step approach, combined with a careful evaluation of the provided options, allows us to confidently arrive at the correct answer. By mastering these techniques, you can tackle similar problems with ease and precision.