Hey guys! Let's dive into the fascinating world of functions, specifically focusing on how to determine the range of a function. Today, we're tackling the function f(x) = -2|x + 1|. Understanding function ranges is super important in math because it tells us all the possible output values we can get from a function. So, let's break this down step by step and figure out what the range of this function is. We will explore the properties of absolute value functions and how transformations affect their range. By the end of this article, you'll not only know the answer but also understand the process behind it. So, grab your thinking caps, and let’s get started!
Understanding Absolute Value Functions
Before we jump directly into our function, let's quickly recap what absolute value functions are all about. Remember, the absolute value of a number is its distance from zero, no matter the direction. This means |x| is always non-negative; it's either zero or a positive number. Think of it like this: |5| = 5, and |-5| also equals 5. This fundamental property is crucial for understanding how absolute value functions behave. The basic absolute value function, y = |x|, forms a V-shaped graph with its vertex (the pointy bottom) at the origin (0, 0). The graph extends upwards indefinitely, showing that the output (y-value) is always greater than or equal to zero. This gives the basic absolute value function a range of [0, ∞). Understanding this basic form is key because our function, f(x) = -2|x + 1|, is a transformation of this parent function. We need to consider how the transformations—specifically the shift, stretch, and reflection—affect the range. The transformations change the position and orientation of the basic V-shape, and by understanding these changes, we can accurately determine the new range. Let's move on to analyzing how these transformations play out in our specific function.
Analyzing the Transformations
Now, let's break down the transformations applied to our function, f(x) = -2|x + 1|. There are two key transformations happening here that we need to consider carefully. First, we have the |x + 1| part. This represents a horizontal shift. Specifically, adding 1 inside the absolute value function shifts the entire graph one unit to the left. Think of it as finding the value of x that makes the inside zero; in this case, x = -1, which is our new vertex's x-coordinate. Shifting the graph left or right doesn't change the range, but it's an important part of understanding the overall transformation. The second, and more impactful transformation for our range, is the -2 coefficient outside the absolute value. This does two things: It vertically stretches the graph by a factor of 2, making it steeper, and it reflects the graph across the x-axis due to the negative sign. The reflection is crucial because it flips the V-shape upside down. Instead of opening upwards from the vertex, it now opens downwards. This reflection has a significant impact on the range, as it changes the direction in which the function's output values extend. The vertical stretch by a factor of 2 further emphasizes these changes, making the function decrease more rapidly as we move away from the vertex. By recognizing these transformations—the horizontal shift, vertical stretch, and reflection—we can start to visualize how the graph has changed from the basic y = |x|, and this gives us clues about the range. So, how do these transformations affect the highest and lowest possible values of the function?
Determining the Range
To pinpoint the range of f(x) = -2|x + 1|, let’s think about what these transformations do to the possible output values. We know that the absolute value part, |x + 1|, will always produce a non-negative value (zero or positive). This is a fundamental property of absolute values. Now, when we multiply this non-negative value by -2, we flip the sign. A positive number becomes negative, and zero remains zero. This means that the output of -2|x + 1| will always be either negative or zero. There's no way to get a positive output from this function because of the negative sign. So, the function's values are limited to being zero or less. The maximum value occurs when the absolute value part is zero, which happens when x = -1. At this point, f(-1) = -2|(-1) + 1| = -2 * 0 = 0. This tells us that 0 is the highest value the function can reach. As x moves away from -1 in either direction, the absolute value |x + 1| increases, and multiplying by -2 makes the function's value decrease (become more negative). There's no lower bound; the function can take on increasingly negative values as x gets further from -1. Therefore, the range of the function f(x) = -2|x + 1| includes all real numbers less than or equal to 0. This is often written in interval notation as (-∞, 0]. By systematically considering the transformations and their impact on the output values, we’ve successfully identified the range. Now, let's summarize our findings and provide a clear answer to the question.
Final Answer and Conclusion
Alright, guys, after carefully analyzing the function f(x) = -2|x + 1|, we've nailed down its range. We saw how the absolute value function, |x + 1|, is shifted, stretched, and reflected, and we understood how these transformations impact the possible output values. The key takeaway is that the negative sign in front of the 2 flips the graph upside down, ensuring that the function's values are always zero or negative. The correct answer is that the range of f(x) = -2|x + 1| is all real numbers less than or equal to 0. This corresponds to option B. So, there you have it! We've not only found the answer but also walked through the reasoning behind it. Understanding the transformations and their effects on the range is a powerful tool for analyzing functions. Remember, when you're faced with a function like this, break it down step by step, think about the transformations, and consider how they affect the possible output values. With practice, you'll become a range-finding pro in no time! Keep exploring and keep learning!