Range Of Absolute Value Function G(x) = |x - 12| - 2
The question at hand involves finding the range of the function g(x) = |x - 12| - 2. To effectively determine the range, we must first understand the behavior of the absolute value function and how the transformations applied to it affect its output. The absolute value function, denoted as |x|, returns the non-negative magnitude of a real number x, regardless of its sign. This fundamental property will be pivotal in delineating the possible output values, or the range, of the given function g(x). The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range of g(x) = |x - 12| - 2, we need to analyze how the transformations applied to the basic absolute value function |x| affect its possible output values. The absolute value function |x| itself has a range of [0, ∞), meaning it can output any non-negative number. This is because the absolute value always returns the magnitude of the input, effectively making all outputs non-negative. We will explore this in detail to establish a solid foundation for understanding the range of more complex absolute value functions. Let's delve deeper into how transformations of functions can affect their range, particularly focusing on the absolute value function. The transformations applied to the basic absolute value function |x| in this problem are a horizontal shift and a vertical shift. These transformations will directly impact the range of the function. Understanding the behavior of the absolute value function and the impact of these transformations is critical to accurately determining the range of g(x). The analysis will not only provide the correct range but also enhance the understanding of function transformations in a broader mathematical context.
Understanding the Absolute Value Function
To effectively determine the range of g(x) = |x - 12| - 2, a deep understanding of the absolute value function is crucial. The absolute value function, denoted by |x|, is a fundamental concept in mathematics. It transforms any real number into its non-negative counterpart. In simpler terms, it measures the distance of a number from zero, without considering the direction. This characteristic of the absolute value function is what gives it its distinctive V-shaped graph, with the vertex located at the origin (0, 0). The range of the basic absolute value function |x| is [0, ∞), which means the output is always greater than or equal to zero. This is because the absolute value always returns a non-negative value. Now, let's consider how the function g(x) is derived from the basic absolute value function. g(x) = |x - 12| - 2 involves two transformations applied to |x|: a horizontal shift and a vertical shift. These transformations alter the position of the V-shaped graph, and consequently, affect the range of the function. The term (x - 12) inside the absolute value causes a horizontal shift. Specifically, it shifts the graph 12 units to the right. This means the vertex of the V-shaped graph, which was originally at (0, 0), is now located at (12, 0). A horizontal shift does not affect the range of the function; it only changes the x-coordinate of the vertex. The more significant transformation concerning the range is the vertical shift. The - 2 term outside the absolute value causes the entire graph to shift 2 units downward. This means the vertex of the graph, which was at (12, 0), is now located at (12, -2). This vertical shift directly impacts the minimum y-value that the function can attain, and hence, it significantly influences the range. To further clarify, let's think about what this vertical shift does to the possible output values. Since the absolute value part, |x - 12|, will always produce a non-negative value, the smallest value it can take is 0. This occurs when x = 12. The subtraction of 2 from the entire expression then shifts this minimum value down to -2. This insight is crucial for determining the lower bound of the range.
Analyzing the Transformations of g(x)
To determine the range of the function g(x) = |x - 12| - 2, it's essential to analyze the transformations applied to the basic absolute value function, |x|. These transformations dictate how the graph shifts and stretches, consequently impacting the set of possible output values. The function g(x) undergoes two key transformations: a horizontal shift and a vertical shift. Let's break down each transformation to understand their individual effects. The first transformation is the horizontal shift, represented by the term (x - 12) inside the absolute value. This transformation shifts the graph of |x| horizontally by 12 units to the right. To visualize this, imagine the V-shaped graph of |x|. The vertex, which is the point where the two lines meet, is initially at (0, 0). Replacing x with (x - 12) moves this vertex to (12, 0). It's crucial to recognize that horizontal shifts do not affect the range of the function. The range is determined by the vertical extent of the graph, and a horizontal shift only moves the graph left or right. The second transformation is the vertical shift, represented by the term - 2 outside the absolute value. This transformation shifts the entire graph vertically downwards by 2 units. The vertical shift is the key to understanding the range of g(x). It alters the minimum y-value that the function can achieve. The vertex, which was at (12, 0) after the horizontal shift, now moves to (12, -2). This means the lowest point on the graph of g(x) is at a y-value of -2. Now, let's consider the implications of these transformations on the range. The absolute value part, |x - 12|, will always produce a non-negative value, as we discussed earlier. The smallest possible value for |x - 12| is 0, which occurs when x = 12. The vertical shift of - 2 then subtracts 2 from this minimum value. This results in a minimum output value of -2 for the function g(x). Since |x - 12| can take any non-negative value, subtracting 2 from it means the output of g(x) can be any number greater than or equal to -2. Therefore, the range of g(x) is all y-values that are greater than or equal to -2.
Determining the Range of g(x) = |x - 12| - 2
To definitively establish the range of the function g(x) = |x - 12| - 2, we need to synthesize our understanding of the absolute value function and the effects of the transformations. The range, as a reminder, is the set of all possible output values (y-values) that the function can produce. We know that the absolute value function, |x|, returns non-negative values. This means that |x - 12| will also always be greater than or equal to zero, regardless of the value of x. The expression |x - 12| reaches its minimum value of 0 when x = 12. When x is anything other than 12, the value inside the absolute value will be non-zero, and the absolute value will make it positive. The vertical shift of - 2 in g(x) then subtracts 2 from the output of the absolute value part. This means that the minimum possible value for g(x) is when |x - 12| is 0, which gives us g(x) = 0 - 2 = -2. Therefore, -2 is the lowest value in the range of g(x). Now, let's consider what happens as x moves away from 12. As x gets larger than 12 or smaller than 12, the value of |x - 12| increases. This means that the output of g(x), which is |x - 12| - 2, will also increase. There is no upper bound on how large |x - 12| can become, so there is also no upper bound on how large g(x) can become. This implies that g(x) can take any value greater than or equal to -2. In conclusion, the range of the function g(x) = |x - 12| - 2 includes all real numbers that are greater than or equal to -2. This can be expressed mathematically as {y | y ≥ -2}. This notation indicates the set of all y-values such that y is greater than or equal to -2.
Conclusion: The Range of g(x) and its Implications
In summary, after a detailed analysis of the function g(x) = |x - 12| - 2, we have determined that the range of this function is {y | y ≥ -2}. This means that the function g(x) can output any real number greater than or equal to -2. This conclusion was reached by carefully examining the transformations applied to the basic absolute value function and understanding how these transformations affect the set of possible output values. The absolute value function, |x|, is a cornerstone in understanding this concept. It always returns non-negative values, which is crucial in determining the range of any function involving the absolute value. The transformations, specifically the horizontal and vertical shifts, play a significant role in shaping the function's graph and, consequently, its range. The horizontal shift (x - 12) moves the vertex of the absolute value graph 12 units to the right but does not affect the range. The vertical shift - 2, however, is the key factor determining the range. It shifts the entire graph downwards by 2 units, setting the minimum output value at -2. This minimum value, -2, is the lower bound of the range. Since the absolute value part of the function, |x - 12|, can take any non-negative value, subtracting 2 from it means the output of g(x) can be any number greater than or equal to -2. This understanding of the range of absolute value functions has broader implications in mathematics. It helps in solving inequalities, graphing functions, and understanding the behavior of more complex mathematical models. The ability to analyze transformations and their effects on the range of a function is a valuable skill in various areas of mathematics and its applications. Furthermore, this problem highlights the importance of breaking down complex functions into simpler components. By understanding the basic function |x| and the effects of the transformations, we were able to systematically determine the range of g(x). This approach can be applied to other types of functions and transformations, making it a powerful tool in mathematical analysis. The range {y | y ≥ -2} fully describes the set of possible output values for g(x), providing a complete understanding of the function's behavior in terms of its output.
Therefore, the correct answer is {y | y ≥ -2}.
