Finding The Range Of G(x) = |x-12| - 2
Determining the range of a function is a fundamental concept in mathematics, especially when dealing with absolute value functions. In this article, we will delve into finding the range of the function g(x) = |x-12| - 2. Understanding the behavior of absolute value functions and how transformations affect them is crucial for solving this type of problem. We will explore the properties of absolute value, analyze the given function step-by-step, and arrive at the correct range. By the end of this discussion, you should have a clear understanding of how to determine the range of absolute value functions and similar transformations.
Absolute Value Functions: The Basics
Before we dive into the specifics of the function g(x) = |x-12| - 2, it is essential to understand the basics of absolute value functions. The absolute value of a number is its distance from zero on the number line. Mathematically, the absolute value of x, denoted as |x|, is defined as follows:
|x| = x, if x ≥ 0 |x| = -x, if x < 0
This means that the absolute value of any number is always non-negative. For instance, |5| = 5 and |-5| = 5. The graph of the basic absolute value function f(x) = |x| forms a V-shape, with the vertex at the origin (0,0). The range of f(x) = |x| is [0, ∞) because the output is always greater than or equal to zero.
Understanding these fundamental properties is key to analyzing more complex absolute value functions. The V-shape of the graph and the non-negativity of the output are crucial in determining the range when transformations are applied to the basic function.
Analyzing g(x) = |x-12| - 2: A Step-by-Step Approach
To determine the range of the function g(x) = |x-12| - 2, we need to break down the transformations applied to the basic absolute value function f(x) = |x|. The function g(x) involves two transformations: a horizontal shift and a vertical shift. Let's analyze each transformation step-by-step.
1. Horizontal Shift: |x-12|
The first transformation is the horizontal shift represented by |x-12|. This transformation shifts the graph of f(x) = |x| horizontally. Specifically, replacing x with (x-12) shifts the graph 12 units to the right. This is because the vertex of the absolute value function, which is originally at (0,0), is now at (12,0). The horizontal shift does not affect the range of the function. The range of |x-12| is still [0, ∞) since the absolute value ensures that the output is always non-negative.
2. Vertical Shift: |x-12| - 2
The second transformation is the vertical shift represented by subtracting 2 from the absolute value expression. The function g(x) = |x-12| - 2 takes the graph of |x-12| and shifts it downward by 2 units. This shift directly affects the range of the function. Since the minimum value of |x-12| is 0, subtracting 2 from it results in a minimum value of -2. Therefore, the range of g(x) = |x-12| - 2 will be all y-values greater than or equal to -2.
By analyzing these transformations step-by-step, we can clearly see how each affects the function's graph and, consequently, its range. The horizontal shift changes the vertex's x-coordinate, while the vertical shift alters the minimum value of the function, thereby defining its range.
Determining the Range of g(x) = |x-12| - 2
Based on our step-by-step analysis, we can now confidently determine the range of the function g(x) = |x-12| - 2. We established that the absolute value component, |x-12|, ensures that the output is always non-negative. The subsequent subtraction of 2 shifts the entire graph downward by 2 units. This means the minimum value of the function is -2.
Since the absolute value function can take any non-negative value, subtracting 2 from it means the function g(x) can take any value greater than or equal to -2. In set notation, the range of g(x) is represented as {y | y ≥ -2}.
To further solidify this understanding, consider a few examples:
- If x = 12, then g(12) = |12-12| - 2 = 0 - 2 = -2
- If x = 14, then g(14) = |14-12| - 2 = 2 - 2 = 0
- If x = 10, then g(10) = |10-12| - 2 = 2 - 2 = 0
These examples illustrate that the output of the function is always -2 or greater. Thus, the range of g(x) = |x-12| - 2 is indeed {y | y ≥ -2}. This range includes -2 and extends infinitely upwards, encompassing all real numbers greater than or equal to -2.
Understanding the Options
Now that we have determined the range of the function g(x) = |x-12| - 2, let's evaluate the given options to identify the correct one.
The options provided are:
- A. {y | y > -2}
- B. {y | y ≥ -2}
- C. {y | y > 12}
- D. {y | y ≥ 12}
Based on our analysis, we concluded that the range of g(x) includes all y-values greater than or equal to -2. This means that -2 is included in the range. Let's break down each option:
- Option A: {y | y > -2} This option represents all y-values strictly greater than -2. It does not include -2 itself. Since our function g(x) can equal -2 (when x = 12), this option is incorrect.
- Option B: {y | y ≥ -2} This option represents all y-values greater than or equal to -2. This is exactly what we found the range of g(x) to be. This option is correct.
- Option C: {y | y > 12} This option represents all y-values strictly greater than 12. This is incorrect because the function's values are not limited to being greater than 12.
- Option D: {y | y ≥ 12} This option represents all y-values greater than or equal to 12. Similar to option C, this is incorrect as the function's range starts at -2.
Therefore, the correct option is B, which accurately represents the range of the function g(x) = |x-12| - 2.
Conclusion: The Range of g(x) = |x-12| - 2 is {y | y ≥ -2}
In conclusion, we have successfully determined the range of the function g(x) = |x-12| - 2 through a step-by-step analysis. By understanding the properties of absolute value functions and the effects of horizontal and vertical shifts, we were able to identify the correct range as {y | y ≥ -2}. This process involved:
- Understanding the basic absolute value function f(x) = |x| and its range.
- Analyzing the horizontal shift |x-12| and its impact on the graph.
- Analyzing the vertical shift of -2 and its impact on the range.
- Combining these transformations to determine the overall range of g(x).
- Evaluating the given options and selecting the one that accurately represents the range.
This exercise highlights the importance of breaking down complex functions into simpler transformations to understand their behavior. By applying these principles, you can confidently determine the range of various absolute value functions and similar transformations. The key takeaway is that the range is influenced by vertical shifts and the inherent non-negativity of the absolute value function.
