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The range of a function refers to the set of all possible output values it can produce, based on its domain. Understanding a function's range is crucial as it tells us the extent and limit of its output, helping us predict its behavior in different scenarios.

In the context of our problem, the function range of \( y = f(x) \) is given as \((-\infty, -2] \). This means \( f(x) \) can output any real number that is less than or equal to \(-2\), extending towards negative infinity. However, when we consider the absolute value function applied to \( f(x) \), the picture changes completely since negative values are converted to their positive forms.

• For any negative number \( a \) where \( a \leq -2 \), \(|a|\) becomes \(-a\).
• Thus, the value \(-2\) transforms to \(2\) when using absolute value.
• This transformation converts the original negative range into positive numbers, starting from \(2\) onwards.

Graphing a function involves plotting its output values against its input values on a coordinate plane. This visual representation is helpful for examining the behavior, trends, and key characteristics of functions.

For \( f(x) \) with the range \((-\infty, -2] \), its graph would primarily appear below the x-axis because the output values are negative. However, once we apply the absolute value transformation to create \( y = |f(x)| \), the graph undergoes significant changes.

• Absolute values of negative numbers shift above the x-axis, flipping all negative parts to become positive.
• The portion of the graph at \( y = -2 \) will now be at \( y = 2 \) instead, while extending upward without bound.
• This creates a reflection effect, causing the graph to be entirely in the non-negative region.

By graphing, students can see these transformations visually, reinforcing the understanding of how absolute values impact function graphs.

Algebraic transformations involve changing a function using operations like addition, multiplication, or taking absolute values, as shown in the exercise.

The absolute value transformation is of particular interest because it significantly alters the function's range and appearance. It takes every output from the original function and converts each negative result into a positive one, while leaving non-negative values unaffected. Here's how it plays out:

• If a function \( f(x) \) outputs a range of \((-\infty, -2] \), applying absolute values changes the entire negative set to a positive one, beginning at \(2\).
• This effect is a major shift, transforming potentially infinite negative values into a wide array of positive values.

Through algebraic transformations, concepts like absolute values help simplify complex functions or create new properties. Being familiar with these transformations provides powerful tools for modifying and understanding different mathematical scenarios.