Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Give a short answer to each question. If the range of \(y=f(x)\) is \((-\infty,-2],\) what is the range of \(y=|f(x)| ?\)

Step-by-Step Solution

Answer

The range of \(|f(x)|\) is \([2, fty)\).

Step 1: Understand the Function's Range

The range of a function refers to the set of possible output values. For the function \(y = f(x)\), we are given that the range is \((-fty, -2]\). This means the output values of \(f(x)\) are all less than or equal to -2.

Step 2: Apply Absolute Value Transformation

When we apply the absolute value transformation \(y = |f(x)|\), any negative output values become positive. Therefore, any value \(y\) within \((-fty, -2]\) will be transformed to \([2, fty)\) because the absolute value of a negative number results in its positive counterpart.

Step 3: Determine the New Range

The range of \(|f(x)|\) changes any outputs from negative or zero to positive (or zero, because the absolute value of zero is zero). Since \(|f(x)|\) turns every value in the range \((-fty, -2]\) to \([2, fty)\), this becomes the new range of the function.

Key Concepts

Absolute Value Transformation

Understanding absolute value transformation is crucial in adjusting output values of functions. Absolute value, represented as \(|f(x)|\), makes any negative outputs positive while zero remains unchanged. Essentially, it reflects the graph of negative outputs over the x-axis to positive outputs, leaving positive outputs intact.
For example, if a function's range includes negative values, the transformation will convert these to equivalent positive values. This means any range excluding zero, like \((-\infty, -2])\), changes to \([2, \infty)\).
This transformation is so important for analyzing functions because it adjusts how we interpret the range. It leads to a new set of possible output values that exclude negatives altogether, impacting the overall shape and outcome of the function graph.

Function Transformation

Function transformation involves altering a function's output without changing its input explicitly.
Common transformations include translation, reflection, dilation, and more. In the case of absolute value, we are performing a transformation that reflects negative output values over the x-axis. This affects how we graphically interpret and understand the function.
When you apply a transformation like absolute value, you adjust the range of the function, changing its output dynamics. It's important to visualize how these transformations affect both the function itself and its graphical representation. Knowing how to manipulate transformations helps in solving complex mathematical problems by simplifying the patterns seen in the range and domain.

Output Values

Output values refer to the set of results we get from inputting values into a function. They are determined by the range of the function, which specifies all possible outcomes when evaluating the function.
In the context of absolute value transformations, understanding how output values change is key. When a function's range is initially negative or includes negative numbers, applying absolute value ensures those outputs are transformed into positive values or zero.
For instance, if the function \(y=f(x)\) has a range of \((-\infty, -2]\), its output values are initially all less than or equal to -2. After transforming with the absolute value, these output values become all equal to or greater than 2, such as \([2, \infty)\). Recognizing such changes is essential for thoroughly comprehending how functions behave.