Finding The Range Of G(x) = 3|x-1|-1 A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of functions, specifically focusing on determining the range of the function g(x) = 3|x-1| - 1. This might seem a bit daunting at first, but fear not! We'll break it down step by step, ensuring you grasp the underlying concepts and confidently conquer similar problems in the future. So, buckle up and get ready to expand your mathematical horizons!

Understanding the Absolute Value Function

Before we tackle the main problem, let's refresh our understanding of the absolute value function. The absolute value of a number, denoted by |x|, is its distance from zero on the number line. This means that |x| is always non-negative, regardless of whether x is positive or negative. For instance, |3| = 3 and |-3| = 3. This fundamental property of the absolute value function is crucial in determining the range of our function, g(x). Now, considering the range of absolute value functions, it's essential to remember that the absolute value will always return a non-negative value. This is because it represents the distance from zero, and distance cannot be negative. This intrinsic property plays a pivotal role in shaping the overall range of functions that incorporate absolute values, including our featured function, g(x) = 3|x-1| - 1. We'll see exactly how this impacts the final range as we delve deeper into the analysis. Thinking about the transformation occurring within our function, the |x-1| part shifts the standard absolute value function one unit to the right. This horizontal shift doesn't affect the range directly, but it's an important element in understanding the function's behavior. The absolute value component ensures that the output is never negative, which then sets the stage for the subsequent transformations that determine the ultimate range of g(x). Let's keep this in mind as we move forward and dissect the remaining parts of the function.

Deconstructing g(x) = 3|x-1| - 1

Our function, g(x) = 3|x-1| - 1, is a transformation of the basic absolute value function, |x|. To understand its range, we need to analyze how each transformation affects the output values. First, we have the |x-1| term. This represents a horizontal shift of the absolute value function one unit to the right. This shift doesn't affect the range, as the possible output values remain non-negative. Next, we multiply the absolute value term by 3, resulting in 3|x-1|. This is a vertical stretch by a factor of 3. Vertical stretching magnifies the output values, but it doesn't change the fundamental non-negativity. The minimum value remains 0, but all other values are tripled. Finally, we subtract 1 from the entire expression, giving us 3|x-1| - 1. This is a vertical translation downwards by 1 unit. A vertical shift directly impacts the range by shifting all output values by the same amount. In this case, subtracting 1 lowers the entire graph by one unit, which will ultimately affect the minimum value of the range. By carefully considering each of these transformations – the horizontal shift, the vertical stretch, and the vertical translation – we can begin to piece together the behavior of the function and accurately determine its range. We'll see how these transformations interact to define the boundaries of the function's output values.

Finding the Minimum Value

The key to determining the range of g(x) lies in finding its minimum value. Since the absolute value term, |x-1|, is always non-negative, its minimum value is 0. This occurs when x = 1. Now, let's trace how this minimum value propagates through the rest of the function. When |x-1| = 0, the term 3|x-1| also becomes 0. Finally, subtracting 1 gives us 3|x-1| - 1 = -1. Therefore, the minimum value of g(x) is -1. To solidify this understanding, consider what happens when |x-1| takes on values greater than zero. Multiplying any positive value by 3 will result in a larger positive value, and then subtracting 1 will still yield a value greater than -1. This reinforces the idea that -1 is indeed the absolute minimum of the function. We've pinpointed this crucial lower bound, and it's the cornerstone for defining the range. The fact that the absolute value part can never be negative is what makes finding this minimum value so straightforward. As x deviates from 1, the absolute value term increases, and consequently, the entire function value increases as well. This behavior helps us to confidently assert that -1 is the lowest point the function will ever reach.

Determining the Range

We've established that the minimum value of g(x) = 3|x-1| - 1 is -1. Now, we need to determine the upper bound of the range. As |x-1| increases, 3|x-1| also increases, and so does 3|x-1| - 1. There is no upper limit to how large |x-1| can become, as x can take on arbitrarily large positive or negative values. Consequently, g(x) can also become arbitrarily large. This means that the function has no maximum value. Therefore, the range of g(x) includes all values greater than or equal to -1. We can express this range in interval notation as [-1, ∞). This notation clearly indicates that -1 is included in the range (due to the square bracket) and that the range extends infinitely upwards (represented by the infinity symbol and the parenthesis). Think about it this way: as x moves further away from 1 in either direction, the absolute value term |x-1| grows without bound. This growth is amplified by the factor of 3, and even after subtracting 1, the function's value continues to increase indefinitely. This unbounded upward trend is the reason the range extends to positive infinity. By combining our understanding of the minimum value and the unbounded upward behavior, we arrive at the complete range of the function.

The Correct Answer

Based on our analysis, the range of g(x) = 3|x-1| - 1 is [-1, ∞). Therefore, the correct answer is B. [-1, ∞). You nailed it! Understanding the transformations applied to the absolute value function and identifying the minimum value were the keys to solving this problem. This approach can be applied to a wide variety of functions involving absolute values and other transformations. Remember, breaking down the function into its constituent parts and analyzing each part's effect on the output is a powerful strategy in mathematics. By systematically considering the transformations, you can gain a clear picture of the function's behavior and confidently determine its range. Keep practicing these techniques, and you'll become a master of range determination in no time! Always think about how each operation – whether it's a shift, stretch, or reflection – influences the possible output values.

Additional Tips and Tricks

To further solidify your understanding, let's explore some additional tips and tricks for finding the range of functions. Graphing the function is an excellent visual aid. By plotting the graph of g(x) = 3|x-1| - 1, you can clearly see the minimum value and the unbounded upward trend, making the range immediately apparent. Another helpful technique is to consider the end behavior of the function. As x approaches positive or negative infinity, what happens to g(x)? In this case, as x becomes very large (positive or negative), the |x-1| term dominates, causing g(x) to increase without bound. This confirms our earlier conclusion that the range extends to positive infinity. Furthermore, understanding the parent function and its transformations is crucial. The absolute value function, |x|, is the parent function for g(x). By knowing the range of the parent function ([0, ∞)) and how the transformations affect it, you can quickly determine the range of the transformed function. For instance, the vertical shift of -1 in g(x) directly translates the range of the parent function downwards by 1 unit. So, from [0, ∞), it becomes [-1, ∞). Remember, practice makes perfect. The more you work with different functions and their transformations, the more intuitive the process of finding the range will become. Don't hesitate to try out various examples and challenge yourself with more complex scenarios. The key is to break down each problem into manageable steps and apply the principles we've discussed.

Practice Problems

Now that we've thoroughly explored the range of g(x) = 3|x-1| - 1, let's put your newfound skills to the test with some practice problems. These problems will help you solidify your understanding and develop confidence in your ability to tackle similar challenges. Try working through them on your own, and then compare your solutions to the answers provided. This is a great way to reinforce your learning and identify any areas where you might need further review.

  1. What is the range of f(x) = 2|x + 3| + 1?
  2. Determine the range of h(x) = -|x - 2| + 4.
  3. Find the range of k(x) = 0.5|x| - 2.

Answers:

  1. [1, ∞)
  2. (-∞, 4]
  3. [-2, ∞)

Remember, the key to success is to break down each function into its transformations and carefully analyze how each transformation affects the range. Think about the parent function, the shifts, stretches, reflections, and any other operations that are applied. With consistent practice, you'll become a pro at finding the range of all kinds of functions! And don't be discouraged if you encounter challenges along the way. Every problem is an opportunity to learn and grow. Embrace the process, and you'll be amazed at your progress.

Conclusion

Congratulations! You've successfully navigated the intricacies of finding the range of g(x) = 3|x-1| - 1. We've covered the fundamental concepts, explored the transformations, identified the minimum value, and ultimately determined the correct range. By understanding the absolute value function and how transformations affect the output values, you're well-equipped to tackle a wide range of similar problems. Remember, the key takeaways are: the absolute value function is always non-negative, vertical shifts directly impact the range, and identifying the minimum or maximum value is crucial. Keep practicing, keep exploring, and most importantly, keep enjoying the beauty and power of mathematics! The journey of learning is a continuous one, and every step you take expands your understanding and opens doors to new possibilities. So, keep challenging yourself, keep asking questions, and never stop seeking knowledge. The world of mathematics is vast and fascinating, and there's always something new to discover. Embrace the challenges, celebrate the victories, and remember that every problem solved is a step forward on your mathematical journey.