Completing The Table Of Values For F(x) = (x-1)/(x^2-1)

In this article, we will complete the table of values for the function f(x) = (x-1)/(x^2-1). This exercise is a fundamental concept in mathematics, particularly in understanding the behavior of functions near points of discontinuity. By evaluating the function at values close to x = 1, we can gain insights into the function's limit and its graphical representation. This article aims to provide a comprehensive understanding of how to compute the values and interpret the results, making it an invaluable resource for students and enthusiasts alike.

Understanding the Function

Before we dive into completing the table, let's first understand the function f(x) = (x-1)/(x^2-1). This is a rational function, which means it is a ratio of two polynomials. The numerator is x - 1, and the denominator is x^2 - 1. The denominator can be further factored as (x - 1)(x + 1). Thus, the function can be rewritten as:

f(x) = (x - 1) / ((x - 1)(x + 1))

Notice that there is a common factor of (x - 1) in both the numerator and the denominator. We can simplify the function by canceling this common factor, but we must be cautious about the value x = 1, where this simplification is not valid. The simplified function is:

f(x) = 1 / (x + 1), for x ≠ 1

This simplification reveals that the function has a removable discontinuity at x = 1. In other words, the function behaves like 1 / (x + 1) everywhere except at x = 1. At x = 1, the original function is undefined because the denominator becomes zero. This concept is crucial for understanding limits and continuity in calculus.

Table of Values

Now, let's proceed to complete the table of values for the function. The table includes values of x approaching 1 from both sides (less than 1 and greater than 1), as well as the value x = 1 itself. We will calculate the corresponding f(x) values for each given x value.

Calculating f(x) Values

We will use the simplified form of the function, f(x) = 1 / (x + 1), for all values except x = 1. For x = 1, the function is undefined, but we will discuss the limit as x approaches 1.

1. x = 0.8

   f(0.8) = 1 / (0.8 + 1) = 1 / 1.8 ≈ 0.5556

2. x = 0.9

   f(0.9) = 1 / (0.9 + 1) = 1 / 1.9 ≈ 0.5263

3. x = 0.99

   f(0.99) = 1 / (0.99 + 1) = 1 / 1.99 ≈ 0.5025

4. x = 0.999

   f(0.999) = 1 / (0.999 + 1) = 1 / 1.999 ≈ 0.5002

5. x = 1

   At x = 1, the original function f(x) = (x - 1) / (x^2 - 1) is undefined because the denominator becomes zero. However, using the simplified form f(x) = 1 / (x + 1), we can find the limit as x approaches 1:

   lim (x→1) f(x) = lim (x→1) 1 / (x + 1) = 1 / (1 + 1) = 1 / 2 = 0.5

   So, although f(1) is undefined, the function approaches 0.5 as x approaches 1.

6. x = 1.001

   f(1.001) = 1 / (1.001 + 1) = 1 / 2.001 ≈ 0.4998

7. x = 1.01

   f(1.01) = 1 / (1.01 + 1) = 1 / 2.01 ≈ 0.4975

8. x = 1.1

   f(1.1) = 1 / (1.1 + 1) = 1 / 2.1 ≈ 0.4762

Completed Table

Analyzing the Results

From the completed table, we can observe several important points about the function f(x). As x approaches 1 from values less than 1, f(x) approaches 0.5 from above. Similarly, as x approaches 1 from values greater than 1, f(x) approaches 0.5 from below. This behavior illustrates the concept of a limit. The limit of f(x) as x approaches 1 exists and is equal to 0.5, even though f(1) itself is undefined. This is a classic example of a removable discontinuity.

Graphical Interpretation

To further understand the behavior of the function, it's helpful to consider its graph. The graph of f(x) = (x - 1) / (x^2 - 1) is identical to the graph of g(x) = 1 / (x + 1), except for a hole at the point (1, 0.5). The graph of g(x) = 1 / (x + 1) is a hyperbola with a vertical asymptote at x = -1 and a horizontal asymptote at y = 0. The hole at (1, 0.5) indicates the removable discontinuity.

Importance of Limits

The concept of limits is fundamental in calculus and analysis. It allows us to study the behavior of functions near points where they may be undefined or exhibit unusual behavior. The limit helps us understand the function's trend as it approaches a particular point, without necessarily requiring the function to be defined at that point. In this case, even though f(1) is undefined, the limit as x approaches 1 provides valuable information about the function's behavior.

Common Mistakes and How to Avoid Them

When evaluating functions and completing tables of values, several common mistakes can occur. It’s essential to be aware of these pitfalls to ensure accurate results.

Mistake 1: Ignoring Discontinuities

One of the most common errors is failing to recognize and address discontinuities. In the function f(x) = (x - 1) / (x^2 - 1), the discontinuity at x = 1 is crucial. Simply plugging in x = 1 into the original function will result in an undefined expression. Always check for potential discontinuities by identifying values of x that make the denominator zero. When a discontinuity is present, simplify the function if possible and consider the limit as x approaches the point of discontinuity.

Mistake 2: Incorrect Simplification

Simplifying a function can make evaluations easier, but it must be done correctly. In our case, simplifying (x - 1) / (x^2 - 1) to 1 / (x + 1) is valid for all x ≠ 1. However, forgetting the condition x ≠ 1 can lead to errors. Always remember the domain restrictions when simplifying functions.

Mistake 3: Calculation Errors

Simple arithmetic errors can easily occur, especially when dealing with decimal values. Double-check each calculation, and consider using a calculator to minimize mistakes. Pay close attention to the order of operations and the signs of the numbers.

Mistake 4: Misinterpreting Limits

Understanding the concept of a limit is vital. The limit of a function as x approaches a value does not necessarily equal the function's value at that point. In our example, the limit as x approaches 1 is 0.5, but f(1) is undefined. Avoid confusing the limit with the function's value at the point.

Mistake 5: Neglecting the Graphical Interpretation

A graphical representation can provide valuable insights into the function's behavior. Neglecting to visualize the function can lead to a less complete understanding. Sketch the graph of the function or use graphing software to observe its behavior, especially near points of discontinuity.

Real-World Applications

Understanding functions and their behavior, including discontinuities and limits, has numerous real-world applications. Here are a few examples:

Physics

In physics, many phenomena are modeled using functions that may have discontinuities or limits. For example, the behavior of a circuit as a switch is closed or opened can be described using functions with discontinuities. Limits are used to describe the behavior of physical systems as they approach certain conditions, such as the speed of an object approaching the speed of light.

Engineering

Engineers use functions to model various systems and processes. For instance, in control systems, functions describe the response of a system to different inputs. Discontinuities and limits are essential for analyzing the stability and performance of these systems. Civil engineers use functions to model the stress and strain on structures, which can help in designing safe and efficient buildings and bridges.

Economics

Economic models often involve functions that describe supply, demand, and cost. These functions can have discontinuities or limits that represent critical points in the market. For example, a supply function might have a discontinuity at a price point where it becomes profitable to enter the market. Economists use limits to analyze the long-term behavior of economic systems.

Computer Science

In computer science, functions are used extensively in algorithms and data structures. The efficiency of an algorithm can be described using functions that may have limits or discontinuities. For example, the time complexity of a sorting algorithm can be expressed as a function of the number of elements to be sorted. Understanding limits helps in analyzing the performance of algorithms as the input size grows.

Data Analysis

Data analysts use functions to model trends and patterns in data. Discontinuities and limits can help identify anomalies or significant changes in the data. For example, a function modeling sales data might have a discontinuity at the time of a major product launch or marketing campaign.

Conclusion

Completing the table of values for the function f(x) = (x - 1) / (x^2 - 1) has provided valuable insights into its behavior, particularly around the point x = 1. We have seen how simplifying the function and understanding the concept of limits allows us to analyze the function even at points of discontinuity. The limit as x approaches 1 is 0.5, even though f(1) is undefined, demonstrating the importance of limits in understanding function behavior. This exercise underscores the fundamental concepts of functions, discontinuities, and limits, which are essential in calculus and various applications in mathematics, science, and engineering. By carefully calculating and interpreting the values, we gain a deeper understanding of the function's properties and its graphical representation. The ability to work with functions and understand their limits is a crucial skill for anyone studying mathematics or related fields. The completed table and analysis serve as a practical example of these concepts in action, reinforcing the theoretical knowledge with a concrete application.