Range Of F(x) = -1/3|x-1|-2: A Step-by-Step Guide
Hey everyone! Today, we're diving into the fascinating world of functions, specifically focusing on how to determine the range of a function. We'll be dissecting the function f(x) = -1/3|x-1| - 2, a classic example that combines absolute values and transformations. If you've ever felt a bit puzzled about ranges, you're in the right place. Let's break it down step by step and make sure we understand exactly how to find the range of this function. By the end, you'll not only know the answer but also the why behind it. So, grab your thinking caps, and let's get started!
Understanding the Absolute Value Function
Before we jump directly into our function, f(x) = -1/3|x-1| - 2, it's crucial to have a solid grasp of the absolute value function. The absolute value, denoted by |x|, might seem simple, but it's the key to understanding how our function behaves. Essentially, the absolute value of a number is its distance from zero, regardless of direction. Think of it as the magnitude without the sign. So, |5| is 5, and |-5| is also 5. This seemingly small detail has a big impact on the range of functions involving absolute values. The absolute value function always returns a non-negative value; it's either zero or a positive number. This is because distance can't be negative. This foundational understanding is super important because it dictates the possible output values of the |x-1| part of our function. Knowing that the absolute value will always be non-negative, we can then start to analyze how the other transformations in the function affect the overall range. So, with this piece of the puzzle in place, we're ready to move on and see how the rest of the function interacts with this core concept.
Transformations: How They Shape the Range
Now that we're comfortable with the absolute value, let's talk about transformations. Our function, f(x) = -1/3|x-1| - 2, isn't just a plain absolute value; it's been transformed. Transformations are like operations that stretch, compress, reflect, or shift a function's graph, and they drastically affect the range. We have a few key transformations happening here: a horizontal shift, a vertical stretch/compression combined with a reflection, and a vertical shift. Let's break them down one by one. First, the |x-1| part represents a horizontal shift. Specifically, it shifts the basic absolute value function one unit to the right. This shift doesn't affect the range directly, but it changes the x-value where the minimum or maximum occurs. Next, the -1/3 multiplier is where things get interesting. The 1/3 part causes a vertical compression, making the graph wider. The negative sign, however, is crucial: it reflects the graph across the x-axis. This reflection is a game-changer for the range, as it flips the function upside down. Instead of opening upwards, our absolute value function now opens downwards. Finally, the -2 at the end causes a vertical shift, moving the entire graph down by 2 units. This shift directly impacts the range by changing the maximum possible y-value. Understanding these transformations is like having a map to navigate the function's behavior. Each transformation plays a role in shaping the final range, and by identifying them, we can predict how the function's output values will be affected.
Analyzing the Function f(x) = -1/3|x-1| - 2
Alright, let's put all the pieces together and really analyze our function: f(x) = -1/3|x-1| - 2. We know it's an absolute value function that's been transformed, but how do these transformations specifically impact the range? Remember, the range is the set of all possible output values (y-values) that the function can produce. We've already established that |x-1| will always be non-negative. Now, let's see what happens step by step. First, we multiply the absolute value by -1/3. Since |x-1| is always greater than or equal to zero, multiplying it by a negative number, -1/3, makes the entire term less than or equal to zero. So, -1/3|x-1| will always be zero or negative. Think about it: if you take a non-negative number and multiply it by a negative, you'll always end up with a non-positive number. Next, we subtract 2 from this result. If we're starting with a number that's zero or negative and then subtract 2, we're shifting the entire set of possible values down by 2 units. This means that the maximum possible value of the function will be when -1/3|x-1| is equal to zero (which happens when x = 1). In this case, f(x) = 0 - 2 = -2. For any other value of x, the |x-1| term will be positive, making -1/3|x-1| negative, and thus f(x) will be less than -2. This is the core insight: the function's output values will never be greater than -2. They can be equal to -2, but they will always be less than or equal to -2. By carefully tracing the effects of each transformation, we've pinpointed the upper limit of our range. This kind of step-by-step analysis is crucial for tackling more complex functions and understanding their behavior.
Determining the Range: Step-by-Step
Let's nail down the range of f(x) = -1/3|x-1| - 2 with a clear, step-by-step approach. This is super helpful for any function, so pay close attention! First, we need to consider the base function, which is the absolute value, |x|. We know that |x| is always greater than or equal to zero. This is our starting point. Now, let's incorporate the horizontal shift. The |x-1| term shifts the graph one unit to the right, but it doesn't change the fundamental fact that the expression is still greater than or equal to zero. So, |x-1| >= 0. Next, we deal with the vertical compression and reflection: -1/3|x-1|. Multiplying by -1/3 flips the inequality because we're multiplying by a negative number. So, now we have -1/3|x-1| <= 0. This tells us that this part of the function will always be zero or negative. Finally, we apply the vertical shift: -1/3|x-1| - 2. We're subtracting 2 from an expression that's already zero or negative. This means the entire expression will be less than or equal to -2. In mathematical notation, we can write this as f(x) <= -2. So, what does this mean for the range? It means that the function's output values can be any real number that is less than or equal to -2. There's no lower bound; the function can go as far down as negative infinity. The key takeaway here is the systematic approach. By breaking down the function into its components and analyzing how each transformation affects the possible output values, we can confidently determine the range. This method isn't just for this specific function; it's a powerful tool for analyzing a wide variety of functions.
The Final Answer and Why
Okay, guys, we've done the work, and now we're ready to state the range of the function f(x) = -1/3|x-1| - 2. Based on our analysis, we know that the function's output values are always less than or equal to -2. This means the range includes -2 and all real numbers less than -2. So, the correct answer is B. all real numbers less than or equal to -2. But let's really understand why this is the answer. It's not just about memorizing steps; it's about understanding the underlying principles. The absolute value part, |x-1|, ensures that we always start with a non-negative value. The -1/3 does two crucial things: it compresses the graph vertically, making it wider, and more importantly, it reflects the graph across the x-axis. This reflection is what turns the function upside down, so instead of having a minimum value, it now has a maximum value. Finally, the -2 shifts the entire graph down by 2 units. This shift dictates the maximum possible y-value, which is -2. Because the function is flipped upside down, all other y-values will be below this maximum. Thinking about it this way, we can see that the range is a direct consequence of these transformations. Each part of the function plays a role in shaping the set of possible output values. So, when you approach a similar problem, remember to break it down, analyze the transformations, and think about how they affect the function's behavior. With practice, you'll be able to determine the range of even the trickiest functions with confidence!
Common Mistakes to Avoid
When tackling range problems, especially those involving absolute values and transformations like in our function f(x) = -1/3|x-1| - 2, it's easy to stumble into common pitfalls. Let's highlight some of these so you can steer clear! One frequent mistake is overlooking the impact of the negative sign in front of the 1/3. Many people correctly identify that the absolute value will produce non-negative values, but they forget that multiplying by a negative number flips the sign. This reflection across the x-axis is crucial, and forgetting it can lead you to think the function has a minimum value instead of a maximum. Another mistake is misinterpreting the vertical shift. While it's clear that subtracting 2 shifts the graph down, some might not fully grasp that this directly changes the maximum possible value in the range. They might get confused and think it only affects the lower bound. It's also common to make errors if you try to memorize rules without understanding the underlying concepts. For instance, you might remember that absolute value functions usually have a range of y >= something, but applying this blindly without considering the reflection can lead to the wrong answer. A purely rote approach doesn't work here; you need to understand why the transformations are doing what they're doing. Finally, another pitfall is not visualizing the graph. A quick sketch, even a rough one, can often clarify the function's behavior and make the range much more apparent. If you can see the graph opening downwards with a maximum at y = -2, the range becomes immediately obvious. So, to avoid these mistakes, always remember to consider the sign, understand the vertical shift's impact on the maximum value, avoid rote memorization, and don't underestimate the power of visualization. By keeping these points in mind, you'll be well-equipped to conquer range problems with confidence!
Practice Problems
To really solidify your understanding of finding the range of functions, especially those with absolute values and transformations like f(x) = -1/3|x-1| - 2, practice is key! Let's look at a few more examples to get you warmed up. These will help you apply the concepts we've discussed and identify any areas where you might need more clarification. Try working through these on your own, and then compare your approach and answers with the explanations. This active learning is the best way to truly master the skill. Here are a few practice problems:
- What is the range of the function
g(x) = 2|x + 3| - 1? - Determine the range of
h(x) = -|x - 2| + 4. - Find the range of
k(x) = -2|x + 1| - 3.
For each of these, follow the same steps we used for our original function. Identify the transformations, think about how they affect the absolute value's output, and then determine the possible y-values. Remember to pay close attention to reflections (negative signs) and vertical shifts. After you've given these a shot, think about other variations you could encounter. What if there was a horizontal stretch or compression? What if there were multiple transformations combined? The more you practice and experiment, the more comfortable you'll become with these types of problems. And don't be afraid to use graphing tools to visualize the functions – it's a fantastic way to check your work and build your intuition. So, go ahead, dive into these practice problems, and keep honing your skills. You've got this!
Conclusion
We've journeyed through the process of finding the range of the function f(x) = -1/3|x-1| - 2, and hopefully, you've gained a solid understanding of how to tackle similar problems. Remember, it's not just about getting the right answer; it's about understanding why the answer is what it is. We started by revisiting the absolute value function and its fundamental property of always producing non-negative values. Then, we delved into transformations – horizontal shifts, vertical stretches/compressions, reflections, and vertical shifts – and how they shape the function's graph and, crucially, its range. We analyzed our specific function step by step, carefully tracing the effects of each transformation. We established a systematic approach for determining the range, and we highlighted common mistakes to avoid, like overlooking the impact of reflections or relying on rote memorization. Finally, we emphasized the importance of practice and provided some additional problems to help you solidify your skills. The key takeaway is that finding the range is a process of understanding the function's behavior. By breaking it down into its components and analyzing the transformations, you can confidently determine the set of all possible output values. So, keep practicing, keep visualizing, and keep asking why. With a solid grasp of these concepts, you'll be well-equipped to conquer the range of any function that comes your way! Keep up the great work, everyone!
