Range Of G(x) = |x - 12| - 2 A Comprehensive Guide

In mathematics, understanding the range of a function is crucial for grasping its behavior and characteristics. The range represents the set of all possible output values (y-values) that a function can produce. In this article, we will delve into the function g(x) = |x - 12| - 2 and determine its range. This involves analyzing the function's structure, considering the properties of absolute values, and identifying any restrictions on the output values. By the end of this exploration, you will have a clear understanding of how to find the range of this particular function and the mathematical principles behind it.

The function g(x) = |x - 12| - 2 is a transformation of the absolute value function. To understand it fully, let's break it down step by step. The core of the function is the absolute value expression, |x - 12|. The absolute value of any real number is its distance from zero, which is always non-negative. This means that |x - 12| will always be greater than or equal to zero. The transformation x - 12 inside the absolute value shifts the standard absolute value function 12 units to the right along the x-axis. This shift doesn't affect the range, as the minimum value remains zero.

Next, we subtract 2 from the absolute value expression. This vertical shift moves the entire function down by 2 units along the y-axis. The minimum value of |x - 12| is 0, so the minimum value of |x - 12| - 2 is 0 - 2 = -2. Since the absolute value is always non-negative, subtracting 2 from it means the output will always be greater than or equal to -2. This understanding of the function's transformations and the properties of absolute values is key to determining its range.

Analyzing the Absolute Value Component |x - 12|

To dissect the function g(x) = |x - 12| - 2, a closer examination of the absolute value component, |x - 12|, is essential. The absolute value function, denoted by |x|, returns the non-negative magnitude of a real number x, irrespective of its sign. This means |x| = x if x is non-negative, and |x| = -x if x is negative. This fundamental property ensures that the output of an absolute value function is always zero or positive.

In our case, we have |x - 12|, which represents the distance between x and 12 on the number line. This expression will be zero when x equals 12, and it will increase as x moves away from 12 in either direction. Crucially, |x - 12| can never be negative, as it measures a distance. The minimum value of |x - 12| is 0, occurring when x is 12. As x deviates from 12, the value of |x - 12| grows without bound. For instance, if x is 10, |10 - 12| = |-2| = 2. If x is 15, |15 - 12| = |3| = 3. This behavior is symmetrical around x = 12, reflecting the nature of absolute values.

Understanding this behavior is paramount for determining the range of the overall function. The absolute value component sets a lower limit for the function's output, which is then further modified by the subtraction of 2. This careful dissection of the absolute value term lays the groundwork for a comprehensive understanding of the function's range.

The Impact of Subtracting 2

The final step in understanding the function g(x) = |x - 12| - 2 involves analyzing the effect of subtracting 2 from the absolute value expression, |x - 12|. As we've established, the absolute value term |x - 12| is always non-negative, with a minimum value of 0 when x equals 12. Subtracting 2 from this expression shifts the entire graph of the function vertically downwards by 2 units. This transformation directly affects the range of the function, as it lowers the minimum possible output value.

When |x - 12| is at its minimum, which is 0, the value of g(x) becomes 0 - 2 = -2. This means that -2 is the lowest possible value that g(x) can attain. As |x - 12| increases (as x moves away from 12), the value of g(x) will also increase. Since |x - 12| can grow indefinitely, so can g(x). There is no upper bound to the values that g(x) can take. For example, if |x - 12| is 5, then g(x) is 5 - 2 = 3. If |x - 12| is 10, then g(x) is 10 - 2 = 8. This demonstrates that g(x) can take on any value greater than or equal to -2.

This subtraction of 2 is critical in defining the range of the function. It sets the lower bound of the range, ensuring that no output value will be less than -2. The combination of the absolute value and the subtraction creates a V-shaped graph that opens upwards, with its vertex at the point (12, -2). Understanding this vertical shift is key to accurately identifying the range of the function.

Having analyzed the function g(x) = |x - 12| - 2 in detail, we can now confidently determine its range. Recall that the range of a function is the set of all possible output values. We've established that the absolute value term, |x - 12|, is always non-negative, meaning it is greater than or equal to 0. The minimum value of |x - 12| occurs when x = 12, where the value is 0.

When we subtract 2 from |x - 12|, we shift the entire function down by 2 units. This means that the minimum value of g(x) is 0 - 2 = -2. Since |x - 12| can take on any non-negative value, g(x) can take on any value that is -2 or greater. There is no upper limit to the values g(x) can achieve because as x moves further away from 12, |x - 12| increases without bound, and consequently, g(x) also increases without bound.

Therefore, the range of the function g(x) = |x - 12| - 2 is the set of all real numbers greater than or equal to -2. In set notation, this is represented as {y | y ≥ -2}. This means that the output of the function can be -2, or any number larger than -2, but it cannot be any number less than -2. This range reflects the vertical shift caused by subtracting 2 and the fundamental properties of the absolute value function.

Expressing the Range in Different Notations

To express the range of the function g(x) = |x - 12| - 2, which we have determined to be all real numbers greater than or equal to -2, several notations can be used. Each notation provides a slightly different way of representing the same set of values, and understanding these different representations is crucial for mathematical literacy.

1. Set Notation: As previously mentioned, set notation provides a concise and formal way to define the range. The range can be expressed as {y | y ≥ -2}. This notation reads as