Range Of G(x) = 3|x-1| - 1 How To Find It

by ADMIN 42 views

Hey guys! Let's dive into the fascinating world of functions and explore how to determine the range of a function. In this article, we're going to break down the function g(x) = 3|x-1| - 1. Specifically, we'll figure out the range of this absolute value function step-by-step, making sure we understand every twist and turn. Understanding the range of functions like this is not just a mathematical exercise; it's a fundamental concept that helps us grasp how functions behave and what values they can produce. So, let's get started and unlock the secrets behind this function's range!

Understanding the Absolute Value Function

Before we jump into the specifics of g(x), let's take a moment to really understand the absolute value function. The absolute value, denoted by |x|, is essentially the distance of a number from zero. Think of it as a number's magnitude, stripping away its sign. So, whether you have a positive number or a negative number, the absolute value always gives you a non-negative result. For example, |3| is 3, and |-3| is also 3. This key characteristic of the absolute value function—that it always returns a non-negative value—is crucial in determining the range of functions that involve absolute values. When we're dealing with a function like g(x) = 3|x-1| - 1, the absolute value part, |x-1|, ensures that whatever value we get inside the absolute value bars, the result will be non-negative. This non-negativity is what shapes the overall behavior and the possible output values (the range) of the function. Understanding this basic principle is the first step in unraveling the range of our function. We need to consider how this non-negative nature interacts with other operations in the function, like the multiplication by 3 and the subtraction of 1, to finally determine the set of all possible output values.

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

To really nail down the range of g(x) = 3|x-1| - 1, we need to break down the function piece by piece and see how each part contributes to the final output. Let's start with the absolute value part, |x-1|. As we discussed, the absolute value always gives us a non-negative result. This means |x-1| will always be greater than or equal to zero, regardless of the value of x. Now, let's consider the next operation: the multiplication by 3. When we multiply a non-negative value by 3, we still get a non-negative value. So, 3|x-1| will also be greater than or equal to zero. This is a crucial step because it sets the foundation for our range. Finally, we have the subtraction of 1. This is where things get interesting. Subtracting 1 from 3|x-1| shifts the entire range downwards. Since 3|x-1| is always greater than or equal to zero, 3|x-1| - 1 will be greater than or equal to -1. This means the minimum value that g(x) can take is -1. But what about the maximum value? Since |x-1| can grow infinitely large as x moves away from 1, 3|x-1| can also grow infinitely large. And when we subtract 1, it still remains infinitely large. This tells us that there's no upper bound to the values g(x) can take. Therefore, by carefully analyzing each operation in the function, we can start to visualize the range: it starts at -1 and extends infinitely upwards. This step-by-step approach is key to understanding how different parts of a function influence its overall behavior and its range.

Determining the Range

Alright, let's get down to business and pinpoint the range of g(x) = 3|x-1| - 1. We've already laid the groundwork by understanding the absolute value function and analyzing how each operation in g(x) affects the output. We know that the absolute value part, |x-1|, ensures the result is always non-negative. This non-negativity is then amplified by the multiplication by 3, keeping the result non-negative. The final piece of the puzzle is the subtraction of 1, which shifts the entire range downwards. We've established that the minimum value of g(x) occurs when |x-1| is at its minimum, which is 0 (when x = 1). Plugging this into our function, we get g(1) = 3|1-1| - 1 = -1. So, -1 is the lowest value in our range. Now, let's think about the upper bound. As x moves further away from 1, |x-1| increases, and so does 3|x-1|. Subtracting 1 doesn't change the fact that this value can grow infinitely large. This means there's no limit to how high g(x) can go. It can take on any value greater than or equal to -1. So, putting it all together, the range of g(x) includes all real numbers greater than or equal to -1. In interval notation, we express this as [-1, ∞). The square bracket indicates that -1 is included in the range, and the infinity symbol with the parenthesis indicates that there's no upper limit. Understanding how the minimum and maximum values are determined by the function's structure is crucial in accurately identifying the range. This comprehensive analysis ensures we're not just guessing but truly understanding the function's behavior.

The Correct Answer and Why

So, after our deep dive into g(x) = 3|x-1| - 1, we've nailed down the range. We know that the function's output values start at -1 and extend infinitely upwards. Looking at the options provided, the one that perfectly matches our findings is B. [-1, ∞). Let's quickly break down why the other options don't fit the bill:

  • A. (-∞, 1]: This option suggests the range includes all values less than or equal to 1. However, our function's output is always greater than or equal to -1, so this option is incorrect.
  • C. [1, ∞): This option indicates the range includes all values greater than or equal to 1. While the function can certainly produce values in this range, it misses the crucial part between -1 and 1, making it an incomplete answer.
  • D. (-∞, ∞): This option suggests the function can take any real number as its output. While absolute value functions can have a wide range, the -1 in our function acts as a lower bound, preventing the output from going below -1. So, this option overestimates the range.

By carefully analyzing the function and understanding how its components interact, we've confidently arrived at the correct answer: B. [-1, ∞). This not only gives us the right solution but also reinforces our understanding of how to determine the range of a function. Each step of our analysis, from understanding the absolute value to considering the transformations, plays a crucial role in accurately identifying the range.

Visualizing the Range

To really solidify our understanding of the range of g(x) = 3|x-1| - 1, let's take a moment to visualize it. Graphing the function can provide a clear picture of its behavior and the values it can take. If you were to plot this function on a graph, you'd see a V-shaped curve. This shape is characteristic of absolute value functions. The vertex (the bottom point of the V) is located at the point (1, -1). This is because the minimum value of |x-1| is 0, which occurs when x = 1, and thus the minimum value of g(x) is 3(0) - 1 = -1. The V-shape extends upwards from this point, indicating that the function's values increase as x moves away from 1 in either direction. The left side of the V goes up and to the left and the right side of the V goes up and to the right. This visual representation perfectly aligns with our algebraic analysis. We determined that the range starts at -1 and extends infinitely upwards, and the graph clearly shows this. There are no values of g(x) below -1, and the function continues to increase without bound as we move along the x-axis. Visualizing the range not only confirms our calculations but also gives us a more intuitive understanding of the function's behavior. It's like seeing the answer in action, reinforcing the connection between the equation and its graphical representation. This holistic approach—combining algebraic analysis with visual interpretation—is a powerful way to master the concept of function ranges.

Key Takeaways for Finding the Range

Before we wrap up our exploration of the range of g(x) = 3|x-1| - 1, let's recap some key takeaways that can help you tackle similar problems in the future. These strategies are not just about getting the right answer; they're about building a solid understanding of how functions work. First and foremost, always start by understanding the basic functions involved. In our case, it was the absolute value function. Knowing that |x| always returns a non-negative value is crucial. Next, break down the function step-by-step. Analyze how each operation (multiplication, subtraction, addition, etc.) affects the output. This helps you understand how the function transforms the input values. Identify any minimum or maximum values. These often determine the boundaries of the range. In our example, the subtraction of 1 set the lower bound for the range. Don't hesitate to visualize the function. Sketching a graph, even a rough one, can give you a clear picture of its behavior and confirm your algebraic analysis. Finally, practice, practice, practice! The more you work with different functions, the better you'll become at recognizing patterns and applying these strategies. Remember, finding the range is not just a mathematical exercise; it's about understanding the behavior of functions and how they map input values to output values. These takeaways will serve as your toolkit for confidently approaching range-finding problems.

By understanding the properties of absolute value functions and systematically analyzing the given function, we correctly identified the range of g(x) = 3|x-1| - 1 as [-1, ∞). Remember, the key is to break down the function, understand the impact of each operation, and visualize the result. Keep practicing, and you'll become a pro at determining the range of any function!