Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Graph each finction in the standand viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. $$f(x)=-|x|:(0, \infty)$$

Step-by-Step Solution

Answer

The function is decreasing on the interval \((0, \infty)\).

Step 1: Understand the Function

The function given is \( f(x) = -|x| \). The vertical bars represent the absolute value, and the negative sign indicates that the absolute value is flipped upside down.

Step 2: Identify the Interval

The specified interval for the function is \((0, \infty)\). This means we only consider the domain of the function for values of \(x\) greater than 0.

Step 3: Analyze the Effect of Negative Absolute Value

The function \( f(x) = -|x| \) will create a V-shaped graph, but since it's negative, the central point will be a maximum at \(x = 0\) and the arms will extend downward.

Step 4: Graph the Function on a Calculator

Using a graphing calculator, input the function \( f(x) = -|x| \) and set the window to standard viewing. Observe how the graph behaves from \(x = 0\) to \(x = \infty\).

Step 5: Trace and Describe the Behavior

Start tracing the graph from a point slightly greater than 0. As you move to the right along the positive x-axis, observe the path of the graph.

Step 6: Determine if the Function is Increasing or Decreasing

While tracing from left to right starting from \(x > 0\), the graph consistently moves downward. This indicates that the function is decreasing along the interval \((0, \infty)\).

Key Concepts

Absolute value functions

Absolute value functions play a crucial role in understanding the behavior of certain mathematical expressions. The absolute value of a number, denoted by vertical bars around the number (like \(|x|\)), represents its distance from zero on the number line. Whether the input is negative or positive, the output will be non-negative.
In the context of functions, when you apply absolute value to an entire function (e.g., \(f(x) = |x|\)), it essentially "folds" the graph along the x-axis.
This folding effect creates a V-shape because negative y-values are reflected across the x-axis to become positive. Consequently, absolute value functions are always non-negative, except when additional transformations are applied.

• If the function contains negative coefficients, like in \(-|x|\), it flips the typical V-shape upside down, making what’s typically a minimum point at the vertex into a maximum.
• This transformation affects how we interpret the graph’s movement and adds complexity to its behavior.

Understanding these transformations is key when graphing or analyzing absolute value functions.

Behavior of functions

The behavior of functions involves understanding how a function acts over a domain or interval. With absolute value functions, especially when modified with additional operations like negation, the behavior becomes more nuanced.
For the function \(f(x) = -|x|\), the initial behavior can be discerned from its basic transformation. Here, negating the absolute value function means every outgoing line from the vertex (typically at the origin for \(|x|\) functions) dips downwards, indicating an inverted V-shape.

• The function's fundamental behavior involves a constant descent along its defined intervals.
• This departure from a standard upward V-shape into a downward slope means that the function doesn't have increasing segments along this acknowledged interval of \((0, \infty)\).

Recognizing these trends by examining initial domains and subsequent graph outputs is a critical skill in identifying function behavior.

Decreasing functions

A function is deemed decreasing over an interval when, as you move from left to right, the function values consistently drop. This decrease is visually represented by a downward slope on the graph.
In the case of the function \(f(x) = -|x|\), the negative absolute value transformation results in the function decreasing as \(x\) progresses through the interval \((0, \infty)\).

• Every point to the right of the origin corresponds to a lower y-value than the point preceding it.
• This strict declining trend without any upticks ensures that at every point, the function exhibits a decreasing nature.

Understanding decreasing behavior in functions is crucial for analyzing and predicting future outputs, even for complex expressions.