Problem 27 - 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 __________. $$f(x)=-|x| ;(-\infty, 0)$$

Step-by-Step Solution

Answer

Over the interval \((- fty, 0)\), the function is increasing.

Step 1: Understand the Function

The function given is \( f(x) = -|x| \). Since \(|x|\) represents the absolute value of \(x\), \( f(x) \) will flip every positive output of \( |x| \) to its negative, making the graph of \( f(x) \) an inverted "V" shape.

Step 2: Identify the Interval

The interval we are considering is \((-fty, 0)\), which means we are examining the behavior of the function only for negative values of \(x\).

Step 3: Analyze Function Behavior

For negative values of \(x\), \(|x| = -x\). This means \( f(x) = -(-x) = x \). Therefore, within the interval \((-fty, 0)\), the function behaves like \(x\) which implies it's a linear function with a positive slope.

Step 4: Observe Graph Behavior

Since \( f(x) = x \) within our interval \((-fty, 0)\), as we move from left to right (approaching zero from the negative side), the function is increasing.

Step 5: Fill in the Blank

Over the interval \((-fty, 0)\), the function \( f(x) = -|x| \) is increasing as it is equivalent to \( f(x) = x\) in this interval.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by the symbol \(|x|\). This makes \(|x|\) always non-negative. For example, \(|-3| = 3\) and \(|3| = 3\). In essence, it strips the number of its sign.

In the context of the function \(f(x) = -|x|\), the absolute value takes \(x\) and converts any negative values to positive before the negative sign in front of \(f(x)\) reverses it again. This characteristic gives the graph a signature inverted 'V' shape. Understanding absolute value is key for predicting how the \(f(x) = -|x|\) will transform across different intervals.

Interval Notation

Interval notation is a concise way of describing a range of values along a number line. It uses brackets or parentheses to signify whether endpoints are included or excluded.

• Parentheses \( ( ) \) denote that an endpoint is not included.
• Brackets \( [ ] \) indicate that an endpoint is included.

In the problem, the interval \((-\infty, 0)\) signifies that we are looking at all numbers less than zero, excluding zero itself.

Understanding interval notation helps in demarcating which portion of the function's behavior you're interested in analyzing. It’s particularly useful for identifying and describing segments of a function's graph within certain bounds.

Function Behavior

Function behavior describes how a function acts over a set range of inputs—whether it increases or decreases. When considering behavior, especially within a given interval, you focus on how the function's outputs change as you progress from one end of the interval to the other.

For the function \(f(x) = -|x|\) over the interval \((-\infty, 0)\), you see that \(|x| = -x\) for negative \(x\). Therefore, \(f(x) = -(-x) = x\), making not \(f(x)\) a straightforwardly linear function with a positive slope in this interval. Thus, it increases as you move towards zero from negative infinity.

Graph Analysis

Graph analysis involves interpreting the visual behavior of a function. You're examining how the function's graph moves, rotates, and scales across a coordinate plane.

For \(f(x) = -|x|\), the graph typically forms an inverted 'V'. However, considering the specified interval \((-\infty, 0)\), it translates into a rising line from left to right for negative \(x\) values.

• As \(x\) approaches zero, the graph of this function rises, indicating an increase.
• Graph analysis simplifies the understanding of complex functions by visualizing their behavior.

By visualizing the function \(f(x) = -|x|\), you quickly see the increasing pattern over the interval provided.