Function Analysis Exploring F(x) = X(x-1) Domain And Evaluation

This article delves into the intricacies of the function f(x) = x/(x-1), providing a comprehensive analysis suitable for mathematics enthusiasts and students alike. We will explore its evaluation at specific points, discuss domain restrictions, and uncover key characteristics that define its behavior. This function, a rational function, presents a unique opportunity to understand concepts such as asymptotes, discontinuities, and the interplay between algebraic expressions and their graphical representations. Whether you're grappling with function evaluations or seeking a deeper understanding of domain restrictions, this guide aims to illuminate the essential aspects of f(x) = x/(x-1). By dissecting this function, we'll not only solve specific problems but also build a solid foundation for tackling more complex mathematical challenges. Our journey will begin with a straightforward evaluation of the function at given points, followed by a critical examination of its domain and the values it can take.

Evaluating f(x) at Specific Points

The cornerstone of understanding any function lies in its evaluation at specific points. By substituting various values for x, we can begin to grasp the function's behavior and how it transforms inputs into outputs. In the case of f(x) = x/(x-1), this process involves replacing x with the given value and simplifying the resulting expression. This seemingly simple act unlocks a wealth of information about the function's nature. For instance, observing the output for positive and negative inputs, as well as values close to critical points, provides clues about the function's graph and its overall trend. Moreover, evaluating the function at specific points allows us to identify any patterns or symmetries that may exist. These patterns can be invaluable in predicting the function's behavior in other regions and in sketching its graph. The process of evaluation is not merely a mechanical exercise; it's an exploratory tool that allows us to connect the abstract algebraic definition of a function to its concrete numerical outputs. In the following sections, we will meticulously evaluate f(x) at x=3 and x=-3, demonstrating the application of this principle and highlighting the importance of careful substitution and simplification. This exploration will lay the groundwork for a deeper understanding of the function's domain and its limitations.

(a)(i) Finding the Value of f(3)

Let's embark on our exploration by determining the value of f(3). This involves substituting x = 3 into the function f(x) = x/(x-1). The substitution is a fundamental operation in mathematics, and it's crucial to perform it accurately to arrive at the correct result. In this instance, we replace every instance of x in the function's formula with the number 3. This yields the expression f(3) = 3/(3-1). The next step is to simplify this expression, following the order of operations. First, we perform the subtraction in the denominator: 3 - 1 = 2. This simplifies our expression to f(3) = 3/2. This fraction represents the value of the function when x is equal to 3. It's a specific point on the graph of the function, and it tells us that the function's output is 3/2 when the input is 3. This seemingly simple calculation provides valuable insight into the function's behavior. It allows us to visualize the function's graph at this particular point and to compare it with other points we might evaluate. Furthermore, this exercise underscores the importance of careful arithmetic and attention to detail in mathematical calculations. A small error in substitution or simplification can lead to a drastically different result. Therefore, meticulousness is a key virtue in mathematical problem-solving.

(a)(ii) Determining the Value of f(-3)

Now, let's shift our focus to finding the value of f(-3). This process mirrors the previous calculation but introduces the added consideration of negative numbers. We substitute x = -3 into the function f(x) = x/(x-1), resulting in the expression f(-3) = -3/(-3-1). Here, it's crucial to handle the negative signs with care. We begin by simplifying the denominator: -3 - 1 = -4. This transforms our expression into f(-3) = -3/-4. A negative number divided by another negative number yields a positive result. Therefore, we can simplify this fraction to f(-3) = 3/4. This value represents the function's output when the input is -3. It's another point on the graph of the function, and it provides further information about its overall shape and behavior. The fact that f(-3) is positive, while f(3) was also positive, suggests that the function may have different behaviors in different regions of the x-axis. This observation highlights the importance of evaluating a function at multiple points to gain a comprehensive understanding of its characteristics. Furthermore, this calculation reinforces the importance of mastering arithmetic with negative numbers. A firm grasp of these fundamental principles is essential for success in more advanced mathematical concepts.

Identifying Values Excluded from the Domain of f

The domain of a function is a critical concept in mathematics. It defines the set of all possible input values (x-values) for which the function is defined and produces a valid output. In simpler terms, it tells us what values we can "plug in" to the function without encountering any mathematical errors or undefined results. For the function f(x) = x/(x-1), we must consider the potential for division by zero, a mathematical taboo. Division by zero is undefined because it violates the fundamental principles of arithmetic. Therefore, any value of x that would make the denominator of our function equal to zero must be excluded from the domain. In this case, the denominator is x - 1. Setting this expression equal to zero, we get the equation x - 1 = 0. Solving for x, we find that x = 1. This means that when x is equal to 1, the denominator becomes zero, and the function is undefined. Consequently, the value x = 1 must be excluded from the domain of f(x). The domain of the function is all real numbers except for 1. This restriction has significant implications for the graph of the function, as it creates a vertical asymptote at x = 1. Understanding domain restrictions is crucial for accurately analyzing and interpreting functions, as it prevents us from making erroneous calculations or drawing incorrect conclusions.

(b) Stating the Excluded Value(s) of x

To explicitly state the value(s) of x that must be excluded from the domain of f(x) = x/(x-1), we revisit our analysis of the denominator. As we established earlier, the function is undefined when the denominator, x - 1, is equal to zero. Solving the equation x - 1 = 0 yields x = 1. Therefore, the only value of x that must be excluded from the domain is x = 1. This single value represents a critical point for the function, as it signifies a point of discontinuity. At x = 1, the function is not defined, and its graph exhibits a vertical asymptote. This means that as x approaches 1 from either the left or the right, the function's value approaches positive or negative infinity. The exclusion of x = 1 from the domain has a profound impact on the function's overall behavior and its graphical representation. It's a fundamental characteristic that must be considered when analyzing the function's properties. In conclusion, the value x = 1 is the sole value that must be excluded from the domain of f(x) = x/(x-1), and this exclusion is a direct consequence of the prohibition against division by zero.

Conclusion: Mastering Function Analysis

In this comprehensive exploration of the function f(x) = x/(x-1), we've traversed the essential steps of function analysis. We began by meticulously evaluating the function at specific points, namely f(3) and f(-3), demonstrating the fundamental process of substitution and simplification. These evaluations provided concrete numerical values that offer insights into the function's behavior at particular locations. Next, we delved into the crucial concept of the domain, identifying the value x = 1 as the sole exclusion due to the prohibition against division by zero. This exclusion revealed a critical characteristic of the function, a vertical asymptote at x = 1, which significantly influences its graph and behavior. By systematically addressing these aspects – evaluation and domain restriction – we've not only solved the given problem but also cultivated a deeper understanding of function analysis as a whole. This understanding extends beyond the specific function at hand, equipping us with the tools and insights necessary to tackle a wide range of mathematical challenges. The ability to evaluate functions, identify domain restrictions, and interpret the implications of these characteristics is paramount in mathematics and its applications. As we continue our mathematical journey, the principles and techniques explored here will serve as a solid foundation for more advanced concepts and problem-solving endeavors. The exploration of f(x) = x/(x-1) has been a valuable exercise in honing our analytical skills and appreciating the intricate nature of functions.