Problem 14 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The range of \(y=|f(x)|\) is \([0, \infty)\)."

Step 1: Understanding the Given Range

The problem provides the range of the function \(f(x)\) as \([-2, \infty)\), meaning that the function outputs values starting from \(-2\) and extending to positive infinity.

Step 2: Applying Absolute Value Function

When we apply the absolute value function \(|f(x)|\), all negative values in the range are converted to their positive counterparts. This means any \(y\) value less than 0 becomes positive, while non-negative values remain unaffected.

Step 3: Determining the New Range

The initial range \([-2, \infty)\) included negative values starting from \(-2\). With the absolute value applied, \(-2\) becomes \(2\). Thus, the new range of \(|f(x)|\) is \([0, \infty)\) since 0 itself is the smallest possible absolute value.

Key Concepts

Absolute Value Function

The absolute value function is a foundational concept in mathematics. It transforms any number into its non-negative value. This means that if you have a negative number, applying the absolute value turns it into a positive one.

• The absolute value of 5 remains 5.
• The absolute value of -5 becomes 5.

In the context of functions, if we consider a function output as a variable, say \(f(x)\), the absolute value is denoted as \(|f(x)|\). This has the effect of making every output of the function non-negative.Consider the function \(f(x)\) with a range of values: if some of these values are negative, applying the absolute value ensures they are "flipped" to become positive, while positive values and zero remain unchanged.Thus, when the absolute value function is applied, it guarantees that the resulting range consists only of zero or positive numbers. This principle is essential for understanding how the range of a function broadens when absolute values are applied.

Function Transformation

Function transformation involves shifting, stretching, compressing, or flipping the graph of a function. These transformations modify the visual representation of the function, thereby changing its output values.Using the absolute value transformation as an example, the approach varies according to what needs to be done:

• Vertical Shift: Adding or subtracting a number to the function moves it up or down.
• Horizontal Shift: Adding or subtracting inside the function argument moves it left or right.
• Reflection: The absolute value function specifically reflects any negative part of the function into the positive territory.

In our case, when \(f(x)\) has its range from \([-2, \infty)\), applying \(|f(x)|\) reflects the part of the range below zero across the x-axis. Therefore, the result is that negative outputs from \(f(x)\) become their positive counterparts, effectively morphing its range to \([0, \infty)\). Understanding this transformation is vital when predicting the output of function manipulations.

Graphical Interpretation

Graphical interpretation helps visualize mathematical concepts, making it easier to grasp transformations. For functions, this involves looking at their plots or graphs.The graph of a function \(f(x)\) with a range of \([-2, \infty)\) signifies it starts at -2 and goes upwards. When you apply the absolute value, any portion of the graph beneath the x-axis flips over it.

• The segment of the graph below the x-axis is reflected upwards.
• Once flipped, the lowest point on the y-axis changes from -2 to 0.

This graphical shift is critical because it clearly demonstrates how function output values alter with transformations. The initially introduced values from the graph's negative region, once transformed, join the positive range already above the x-axis.Visualizing this flip helps in understanding how the transformation affects the entire function. This change can be seen as all previously negative values now maintain the same distance from the x-axis but on the positive side, confirming the new range as \([0, \infty)\).