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

In the fascinating world of mathematics, understanding the behavior of functions is paramount. Among the diverse types of functions, absolute value functions hold a unique place, often presenting intriguing challenges and applications. This article delves into the intricacies of determining the range of a specific absolute value function, $g(x) = 3|x-1|-1$. We will explore the fundamental properties of absolute value, dissect the given function, and employ a step-by-step approach to unravel its range. By the end of this comprehensive guide, you will possess a clear understanding of how to tackle similar problems and appreciate the elegance inherent in these mathematical constructs.

Understanding Absolute Value Functions

To effectively determine the range of $g(x) = 3|x-1|-1$, we must first grasp the essence of absolute value functions. At its core, the absolute value of a number represents its distance from zero, irrespective of direction. Mathematically, we define the absolute value of $x$, denoted as $|x|$, as follows:

$|x| = x$ if $x ≥ 0$

$|x| = -x$ if $x < 0$

This definition reveals that the absolute value function always yields a non-negative output. This characteristic is crucial in shaping the behavior and, consequently, the range of functions incorporating absolute value expressions. The absolute value function transforms any negative input into its positive counterpart while leaving non-negative inputs unchanged. This symmetrical behavior around zero is a hallmark of absolute value functions and significantly influences their graphical representation and range.

Furthermore, it is essential to recognize the transformations that can be applied to absolute value functions. These transformations, such as vertical stretches, compressions, and translations, play a pivotal role in altering the function's range. In our specific case, the function $g(x) = 3|x-1|-1$ involves a vertical stretch by a factor of 3, a horizontal translation by 1 unit to the right, and a vertical translation by 1 unit downward. Understanding these transformations is key to accurately determining the range of the function.

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

Now, let's turn our attention to the specific function at hand: $g(x) = 3|x-1|-1$. This function is a transformed version of the basic absolute value function, $|x|$. To decipher its range, we need to dissect its components and understand how they interact.

The core element is the absolute value term, $|x-1|$. This expression represents the distance between $x$ and 1. As we established earlier, the absolute value of any quantity is always non-negative. Therefore, $|x-1|$ will always be greater than or equal to zero. This fundamental property sets the stage for determining the overall range of the function.

Next, we observe the multiplication by 3. Multiplying the absolute value term by 3, i.e., $3|x-1|$, stretches the function vertically. This means that the minimum value of $3|x-1|$ will still be 0 (when $x = 1$), but the function will grow three times faster as $x$ moves away from 1 in either direction. This vertical stretch affects the vertical span of the function's graph and, consequently, its range.

Finally, we have the subtraction of 1. Subtracting 1 from $3|x-1|$ shifts the entire function vertically downwards by 1 unit. This vertical translation is crucial because it directly impacts the lower bound of the range. The minimum value of the function will be lowered by 1 unit, which will be a critical factor in determining the final range.

By carefully considering each component of the function, we can begin to piece together a clear picture of its behavior and, ultimately, its range.

Determining the Range Step-by-Step

To rigorously determine the range of $g(x) = 3|x-1|-1$, we will follow a step-by-step approach, building upon our understanding of absolute value functions and the analysis of the given function.

Step 1: Minimum Value of the Absolute Value Term

As we know, the absolute value of any expression is always non-negative. Therefore, the minimum value of $|x-1|$ is 0, which occurs when $x = 1$. This is the foundation for our range determination.

Step 2: Impact of Multiplication by 3

Multiplying the absolute value term by 3, we get $3|x-1|$. Since $|x-1|$ is always greater than or equal to 0, $3|x-1|$ will also be greater than or equal to 0. The minimum value remains 0, but the vertical stretch is now in effect.

Step 3: Effect of Subtracting 1

Subtracting 1 from $3|x-1|$ gives us the complete function, $g(x) = 3|x-1|-1$. This vertical translation shifts the entire function downwards by 1 unit. Therefore, the minimum value of $g(x)$ occurs when $3|x-1|$ is at its minimum (which is 0), resulting in $g(x) = 0 - 1 = -1$.

Step 4: Upper Bound of the Range

As $x$ moves away from 1 in either direction (i.e., as $|x-1|$ increases), the value of $3|x-1|$ also increases without bound. Consequently, $g(x) = 3|x-1|-1$ will also increase without bound. This implies that there is no upper limit to the values that $g(x)$ can take.

Step 5: Expressing the Range

Based on our analysis, the minimum value of $g(x)$ is -1, and there is no upper bound. Therefore, the range of $g(x) = 3|x-1|-1$ is all real numbers greater than or equal to -1. In interval notation, this is expressed as $[-1, ∞)$.

Conclusion

In conclusion, by meticulously dissecting the function $g(x) = 3|x-1|-1$ and applying our understanding of absolute value functions, we have successfully determined its range to be $[-1, ∞)$. This journey highlights the importance of recognizing the transformations applied to a basic function and how these transformations impact its range. The step-by-step approach we employed serves as a valuable framework for tackling similar problems involving absolute value functions. Understanding these concepts not only enhances your mathematical prowess but also opens doors to appreciating the elegance and interconnectedness of mathematical ideas. By mastering the analysis of absolute value functions, you gain a powerful tool for exploring the world of functions and their applications.

Final Answer

The correct answer is B. [-1, ∞)