Problem 20 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ f(x)=\frac{x}{|x|} $$

Step-by-Step Solution

Answer

The graph consists of two horizontal lines: \( y = -1 \) for \( x < 0 \) and \( y = 1 \) for \( x > 0 \), with a gap at \( x = 0 \).

Step 1: Understanding the Function

The function given is \( f(x) = \frac{x}{|x|} \). This function involves the absolute value of \( x \). Note that this function will give different outputs based on the sign of \( x \). Specifically, for any \( x eq 0 \), \( f(x) \) will be 1 if \( x > 0 \) and -1 if \( x < 0 \). Additionally, \( f(x) \) is undefined at \( x = 0 \).

Step 2: Creating a Table of Values

Let's select a set of values for \( x \) around 0, such as \(-3, -2, -1, 0, 1, 2, 3\). We will calculate \( f(x) \) for each of these values: \(-3/|-3| = -1\), \(-2/|-2| = -1\), \(-1/|-1| = -1\), \(0/|0|\) (undefined), \(1/|1| = 1\), \(2/|2| = 1\), \(3/|3| = 1\). This results in the following table: | \( x \) | \( f(x) \) ||------|-------|| -3 | -1 || -2 | -1 || -1 | -1 || 0 | undefined || 1 | 1 || 2 | 1 || 3 | 1 |

Step 3: Sketching the Graph

From the table, we observe that \( f(x) = -1 \) for all \( x < 0 \) and \( f(x) = 1 \) for all \( x > 0 \). This depicts two horizontal lines: one at \( y = -1 \) for negative \( x \) values and one at \( y = 1 \) for positive \( x \) values. At \( x = 0 \), \( f(x) \) is undefined, so there is no point plotted at \( x = 0 \). To sketch the graph, draw a horizontal line at \( y = -1 \) for \( x < 0 \), and a horizontal line at \( y = 1 \) for \( x > 0 \). Make sure to leave a gap at \( x = 0 \) to denote that the function is undefined at this point.

Key Concepts

Piecewise Function

A piecewise function is a function that is defined by multiple sub-functions, each applying to a specific interval of the main function's domain. In our example, the function \( f(x) = \frac{x}{|x|} \) can be considered a piecewise function because it has different expressions depending on the value of \( x \).
Specifically, it can be defined as:

• \( f(x) = 1 \) for \( x > 0 \)
• \( f(x) = -1 \) for \( x < 0 \)

Thus, the piecewise nature of \( f(x) \) dictates the drawing of two separate horizontal lines on its graph. One must segment these lines properly to demonstrate the change in function at \( x = 0 \).
The distinct behavior of piecewise functions makes them crucial in modeling situations where a rule changes over intervals. Whether it be tax rates or tiered pricing models, comprehending this type of function can deepen the understanding of a wide array of real-world phenomena.

Absolute Value Function

Absolute value functions are mathematical expressions that describe the non-negative distance of a number from zero on the number line. For any number \( x \), the absolute value is represented as \( |x| \), which translates to the following rule:

• \( |x| = x \) if \( x \geq 0 \)
• \( |x| = -x \) if \( x < 0 \)

In the function \( f(x) = \frac{x}{|x|} \), the absolute value operation ensures that \( |x| \) is always positive, even if \( x \) itself is negative. This step is critical because it affects how the numerator and denominator interact in the equation: positive \( x \) retains \( x \)'s sign, while negative \( x \) results in \( -x \), changing the division outcome.
Understanding absolute value functions helps visualize situations like measuring distance, where the notion of direction (positive or negative) is not as important as the total length traveled.

Undefined at a Point

In mathematics, a function is said to be undefined at a point if there is no assigned output value for that input. In our function \( f(x) = \frac{x}{|x|} \), the issue arises when \( x = 0 \). The concept of undefined specifically stems from the fact that division by zero is not possible in math, rendering \( \frac{0}{|0|} \) impossible to compute.
This undefined characteristic is crucial when graphing, as it requires discontinuities or gaps in the graph. In the case of \( f(x) \), you would leave a small open circle at \( (0, \) undefined\() \) because the function is not defined at that point.
Recognizing undefined points is important both in theory and application, alerting you to possible adjustments or considerations needed when interpreting the behavior of a function around such points.