91Ó°ÊÓ

A piecewise function is a function composed of multiple sub-functions, each applying to a certain interval of the main function's domain. In the exercise, the function is defined by different expressions depending on the value of x due to the presence of the absolute value in the denominator. For values where \( x > 0 \), the absolute value \(|x|\) becomes \(x\), transforming the function to \( f(x) = \frac{x^2 - 1}{x - 1} = x + 1 \). Conversely, for \( x < 0 \), \(|x|\) translates to \(-x\), turning the function into \( f(x) = \frac{x^2 - 1}{-x - 1} = -x - 1 \). Because of the different expressions in varying intervals, we analyze each part separately to understand the overall behavior of the function at crucial points, particularly at points of discontinuity or where limits need to be evaluated, such as \( x = -1 \).
The key to handling piecewise functions is understanding each sub-function's domain and the transition points where switchovers occur, indicating potential discontinuities.

One-sided limits look at a function's behavior as the variable approaches a point from either the left or the right side. They provide insights into how functions behave at potential points of discontinuity. In this scenario, we estimate the limit of \( f(x) \) as \( x \) approaches \(-1\).

• From the right \((x \to -1^+)\), we consider values like \(-0.9, -0.99, \) and \(-0.999\) and calculate \( f(x) = x + 1 \). The values trend towards \( 0 \).
• From the left \((x \to -1^-)\), we consider values like \(-1.1, -1.01, \) and \(-1.001\) and calculate \( f(x) = -x - 1 \). Here, the values trend towards \(-2\).

As the limits approached from both sides are not equal, the overall limit at \( x = -1 \) does not exist. Understanding one-sided limits is crucial for detecting points of discontinuity, telling us whether a function's limit is well-defined or divergent at particular points.

Discontinuity occurs in a function where the piecewise parts or expressions do not "connect" at certain points, meaning there is a break or jump in values. In this example, the discontinuity appears at \( x = -1 \) because the limits from the left \((-2)\) and the right \((0)\) do not coincide. Hence, \( f(x) \) shows a jump discontinuity at this point.
Discontinuities can have different forms:

• Jump Discontinuity: Occurs when the left and right limits at a point are not equal.
• Point Discontinuity: Occurs when the function is not defined at a point, but the limits are equal.
• Infinite Discontinuity: Occurs when the function approaches infinite values around a point.

Recognizing the type of discontinuity helps in properly describing the function's behavior around that point, as shown by the example of \( f(x) \) where there is a jump discontinuity.

Graphical analysis involves examining the graph of a function to understand its behavior and characteristics visually, such as continuity, limits, and points. In this exercise, analyzing the graph of the piecewise-defined \( f(x) \) confirms our algebraic findings. By plotting \( f(x) = x + 1 \) for \( x > 0 \) and \( f(x) = -x - 1 \) for \( x < 0 \), we observe a visible jump at \( x = -1 \).
Using tools like Zoom and Trace, we can get precise values near \( x = -1 \) and observe how the graph diverges at this point. This provides a great visual confirmation of the behavior derived through algebraic evaluation.
Graphical analysis is particularly useful for:

• Detecting points of discontinuity more visibly than algebraic methods.
• Understanding trends and behavior of a complex or piecewise function.
• Providing intuitive insights that supplement algebraic conclusions.

Through graphical methods, we gain an intuitive grasp of function behavior, ensuring a comprehensive understanding of concepts like limits and discontinuity as seen in the function \( f(x) \).