Range Of F(x) = -2|x+1| Function: A Detailed Explanation

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on determining the range of the function f(x) = -2|x+1|. This might seem a bit daunting at first, but don't worry, we'll break it down step by step, making sure everyone understands the process. So, buckle up and let's get started!

Understanding the Absolute Value Function

Before we can tackle the range of our function, it's crucial to understand the absolute value function itself. The absolute value of a number, denoted by |x|, is its distance from zero on the number line. This means that the absolute value is always non-negative; it's either zero or a positive number. For example, |3| = 3 and |-3| = 3. This seemingly simple concept is the key to unlocking the range of our function. The absolute value function, mathematically represented as |x|, plays a pivotal role in shaping the behavior of f(x) = -2|x+1|. At its core, the absolute value function transforms any input into its non-negative counterpart. Whether the input is positive or negative, the output is always the magnitude of the number, effectively discarding the sign. For instance, |5| equals 5, and |-5| also equals 5. This fundamental property has a cascading effect on the composite function we're analyzing. When we introduce the transformation |x + 1|, we're shifting the standard absolute value function one unit to the left on the x-axis. This shift doesn't alter the fundamental non-negativity of the output; it merely repositions the function's vertex. The result is a V-shaped graph with its minimum point at x = -1. The magic truly happens when we consider the coefficient -2 in front of the absolute value. This coefficient introduces two critical transformations: a vertical stretch by a factor of 2 and a reflection across the x-axis. The vertical stretch amplifies the output values, making the function steeper, while the reflection flips the graph upside down. So, instead of a V-shape opening upwards, we now have an inverted V-shape opening downwards. This reflection is what ultimately dictates that the function's range will be confined to negative values or zero.

The absolute value function's non-negative nature is paramount in determining the range of f(x). Since |x+1| is always greater than or equal to zero, multiplying it by -2 will always result in a value that is less than or equal to zero. This is because multiplying a non-negative number by a negative number yields a non-positive number. The transformation of the absolute value function, |x+1|, plays a pivotal role in understanding the behavior of the given function. The addition of 1 inside the absolute value shifts the graph horizontally by one unit to the left. This shift is crucial because it changes the point at which the absolute value function reaches its minimum. While |x| reaches its minimum at x=0, |x+1| reaches its minimum at x=-1. This horizontal shift doesn't affect the fundamental property that the absolute value is always non-negative, but it does influence the overall shape and position of the graph. The graph of |x+1| is still a V-shaped graph, but the vertex of the V is now located at x=-1 instead of x=0. This means that the function will behave differently for different values of x, and it's important to consider this shift when determining the range of the function.

The Impact of -2 on the Absolute Value

Now, let's consider the multiplication by -2. This is where things get interesting. Multiplying |x+1| by -2 has two significant effects. First, it vertically stretches the graph by a factor of 2. This means that all the y-values of the function are doubled. Second, and perhaps more importantly, it reflects the graph across the x-axis. This means that the positive y-values become negative, and the negative y-values become positive. Because the absolute value |x+1| is always greater than or equal to 0, multiplying it by -2 will always result in a value less than or equal to 0. This is a crucial observation that directly impacts the range of the function. The multiplication by -2 is not just a simple scaling factor; it fundamentally alters the nature of the function. It takes a function that was always non-negative and transforms it into a function that is always non-positive. This reflection across the x-axis is what ultimately confines the range of f(x) to values less than or equal to zero. In other words, the function will never produce a positive output. This is because the absolute value |x+1| is always non-negative, and multiplying a non-negative number by a negative number will always result in a non-positive number. Understanding this transformation is key to understanding the range of the function.

Determining the Range

Considering the properties we've discussed, the range of f(x) = -2|x+1| is all real numbers less than or equal to 0. Here's why: The absolute value |x+1| is always greater than or equal to 0. Multiplying a non-negative value by -2 always results in a value less than or equal to 0. The maximum value of f(x) occurs when |x+1| = 0, which happens when x = -1. In this case, f(-1) = -2|(-1)+1| = -2(0) = 0. As x moves away from -1, |x+1| increases, and f(x) becomes more negative. There is no lower bound on the values that f(x) can take, as the absolute value can become arbitrarily large. Therefore, the function can take on any value less than or equal to 0. This means that the range of f(x) includes all negative real numbers and zero. In mathematical notation, this can be written as (-∞, 0]. The range of a function is the set of all possible output values. In this case, the output values are the values of f(x). We've shown that f(x) can take on any value less than or equal to 0, so the range of f(x) is the set of all real numbers less than or equal to 0.

To solidify our understanding, let's consider some specific examples. When x = -1, f(x) = 0, which is the maximum value of the function. When x = 0, f(x) = -2. When x = -2, f(x) = -2. As we move further away from x = -1, the function becomes more and more negative, demonstrating that there is no lower limit to the range. Understanding the range of a function is crucial in many areas of mathematics and its applications. It allows us to understand the limitations of a function and the possible values that it can output. In this case, knowing that the range of f(x) is all real numbers less than or equal to 0 tells us that the function will never produce a positive output. This information can be valuable in solving equations, graphing functions, and understanding the behavior of mathematical models.

Conclusion

Therefore, the correct answer is B. all real numbers less than or equal to 0. We've walked through the logic step by step, considering the properties of absolute value and the impact of the -2 coefficient. Hopefully, this explanation has clarified how to determine the range of this type of function. Remember, practice makes perfect, so keep exploring different functions and their ranges! Understanding the range of a function is a fundamental concept in mathematics. It allows us to fully understand the behavior of a function and its potential applications. By breaking down the function into its component parts, we can gain a deeper understanding of how each part contributes to the overall behavior of the function. In the case of f(x) = -2|x+1|, understanding the absolute value function and the impact of the -2 coefficient is key to determining the range. The absolute value function ensures that the output is always non-negative, while the -2 coefficient reflects the graph across the x-axis and stretches it vertically. These transformations combine to create a function whose range is all real numbers less than or equal to 0.