Problem 28 - A Graphical Approach to College Algebra

Question

Graph each function in the standard 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. Over the interval specified, this function is __________. $$f(x)=-|x| ;(0, \infty)$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The given function is \( f(x) = -|x| \). This function represents the opposite (or negative) of the absolute value of \( x \). When you graph the function, you will notice that it is an inverted V shape that opens downwards.

Step 2: Identify the Specified Interval

The interval given is \((0, \infty)\). This means we will look at the section of the graph starting from slightly above 0 (positive side of the x-axis) and extending towards infinity.

Step 3: Analyze the Behavior of the Function

Within the interval \((0, \infty)\), the function \( f(x) = -|x| \) is strictly decreasing. As \( x \) becomes larger, the value of the function becomes more negative, indicating the function is moving downward.

Step 4: Fill in the Blank

Using the analysis from the previous step: Over the interval \( (0, \infty) \), this function is **decreasing**.

Key Concepts

Absolute Value Functions

Absolute value functions are mathematical expressions that represent the distance of a number, variable, or expression from zero on the number line. This distance is always non-negative, so the result of an absolute value function is always greater than or equal to zero. The absolute value of a real number \( x \) is denoted by \( |x| \).

When graphing the basic absolute value function \( f(x) = |x| \), the graph forms a "V" shape. This is because the function mirrors positive and negative inputs to the positive y-axis. The vertex of this "V" occurs at the origin (0,0), where the value of \( x \) is zero.

In the case of the function given in the exercise, \( f(x) = -|x| \), the negative sign in front of the absolute value indicates that the graph of the function is flipped over the x-axis, resulting in an inverted "V" shape. This transformation shifts all outputs of the absolute value function to be non-positive, making them less than or equal to zero.

Decreasing Function

A decreasing function is one where the value of the function diminishes as the input increases. For the function \( f(x) = -|x| \), specifically over the interval \((0, \infty)\), it displays this behavior.

Consider this step-by-step:

• At \( x = 0 \), the function's value is \( 0 \).
• As \( x \) moves to any number greater than \( 0 \), the absolute value \( |x| \) increases, but because it is multiplied by -1, making \( f(x) \) more negative.
• Thus, as \( x \) increases, the outputs or \( f(x) \) values decrease.

To identify a decreasing function on a graph, observe that as you move to the right along the x-axis, the y-values (or function values) steadily drop. This characteristic is a significant aspect as it implies the inverse relationship between the input and output values.

Interval Notation

Interval notation is a mathematical notation used to describe a range of values. It tells us where a function is defined or behaves in a specific way. In the given problem, the interval specified is \( (0, \infty) \).

Here's how interval notation works:

• Parentheses \(( )\) indicate that an endpoint is not included, known as an open interval. For example, \((0, \infty)\) means starting just above zero.
• Brackets \([ ]\) suggest that an endpoint is included, known as a closed interval.
• Infinity symbols \( \infty \) and \(-\infty \) are always accompanied by parentheses since infinity is a concept, not a number, and cannot be "included".

Using interval notation helps succinctly communicate the domain or specific behavior of functions over a specified region, and is especially useful when specifying where a function like \( f(x) = -|x| \) is decreasing on the graph.