Finding Vertical Asymptotes Of F(x) = (5x + 5) / (x^2 + X - 2)
This article delves into the process of identifying vertical asymptotes, focusing on the specific function f(x) = (5x + 5) / (x^2 + x - 2). Vertical asymptotes are crucial features of rational functions, providing insights into the function's behavior as x approaches certain values. Understanding how to find them is essential for graphing and analyzing rational functions effectively. We will explore the steps involved in determining these asymptotes, including factoring, simplifying, and identifying the values of x that make the denominator zero while not making the entire function undefined due to cancellations. By the end of this guide, you will have a clear understanding of how to determine vertical asymptotes and their significance in the context of rational functions. The correct answer to the question is C. x = 1 and x = -2. This article will explain exactly how we arrive at that answer, giving you a solid foundation for tackling similar problems in the future. Identifying vertical asymptotes involves a combination of algebraic manipulation and careful analysis, and this guide will walk you through each step in a clear and concise manner. So, let's embark on this mathematical journey and unravel the intricacies of vertical asymptotes!
Understanding Vertical Asymptotes
Vertical asymptotes are vertical lines that a function approaches but never quite reaches. In the context of rational functions, which are functions expressed as a ratio of two polynomials, vertical asymptotes occur at x-values where the denominator of the function equals zero, provided that the numerator does not also equal zero at the same x-value after simplification. This is a crucial concept, as it highlights the points where the function becomes undefined due to division by zero. However, it's not as simple as just finding the zeros of the denominator. We must first ensure that the rational function is simplified. If a factor is common to both the numerator and the denominator, it can be canceled out, potentially removing a discontinuity (a hole) rather than indicating a vertical asymptote. This simplification process is a key step in accurately identifying vertical asymptotes. The presence of vertical asymptotes significantly impacts the graph of the function, causing it to shoot off towards positive or negative infinity as x approaches the asymptote. Thus, understanding how to locate these asymptotes is paramount in sketching the function's graph and comprehending its overall behavior. Furthermore, vertical asymptotes play a critical role in various applications of rational functions, such as in physics, engineering, and economics, where they can represent physical limitations or constraints. To truly master the concept of vertical asymptotes, one must practice identifying them in different rational functions and understand the algebraic manipulations involved. This includes factoring polynomials, simplifying rational expressions, and analyzing the behavior of the function near the potential asymptotes. By developing a strong understanding of these principles, one can confidently tackle even the most complex problems involving rational functions and their asymptotes.
Step-by-Step Solution for f(x) = (5x + 5) / (x^2 + x - 2)
To determine the vertical asymptotes of the function f(x) = (5x + 5) / (x^2 + x - 2), we need to follow a systematic approach. This involves factoring both the numerator and the denominator, simplifying the expression, and then identifying the zeros of the simplified denominator. Each of these steps is crucial in accurately pinpointing the vertical asymptotes. Firstly, factoring the numerator 5x + 5 is straightforward. We can factor out a 5, resulting in 5(x + 1). This simple factorization reveals a key factor that might play a role in simplification later on. Secondly, we turn our attention to the denominator, x^2 + x - 2. This is a quadratic expression, and factoring it requires finding two numbers that multiply to -2 and add to 1. These numbers are 2 and -1, so we can factor the denominator as (x + 2)(x - 1). Now that we have factored both the numerator and the denominator, we can rewrite the function as f(x) = 5(x + 1) / ((x + 2)(x - 1)). The next crucial step is simplification. We look for common factors in the numerator and the denominator that can be canceled out. In this case, there are no common factors, so the function is already in its simplest form. This means that any values of x that make the denominator zero will indeed correspond to vertical asymptotes. Finally, we identify the values of x that make the denominator zero. Setting (x + 2)(x - 1) = 0 gives us two solutions: x = -2 and x = 1. These are the vertical asymptotes of the function. At these values, the function is undefined, and its graph will approach vertical lines. In summary, by carefully factoring, simplifying, and identifying the zeros of the denominator, we have successfully determined the vertical asymptotes of the given function. This process highlights the importance of algebraic manipulation and attention to detail in analyzing rational functions.
Factoring the Numerator and Denominator
Factoring is a fundamental step in determining the vertical asymptotes of a rational function. This process involves breaking down the numerator and the denominator into their simplest multiplicative components. For our function, f(x) = (5x + 5) / (x^2 + x - 2), we begin by factoring the numerator, 5x + 5. This is a straightforward process as we can factor out the common factor of 5, yielding 5(x + 1). This factorization is crucial as it reveals the factor (x + 1), which might potentially cancel out with a similar factor in the denominator, thereby simplifying the function. Next, we tackle the denominator, which is the quadratic expression x^2 + x - 2. Factoring a quadratic expression involves finding two numbers that multiply to the constant term (-2) and add up to the coefficient of the linear term (1). In this case, the numbers are 2 and -1, as 2 * -1 = -2 and 2 + (-1) = 1. Thus, we can factor the denominator as (x + 2)(x - 1). This factorization is critical as it exposes the potential values of x that could make the denominator zero, which are the key candidates for vertical asymptotes. Factoring both the numerator and the denominator is not merely a mechanical step; it provides a deeper understanding of the function's structure. By expressing the function in its factored form, we can easily identify common factors that can be canceled out, leading to simplification and a more accurate determination of the vertical asymptotes. Moreover, factoring helps us visualize the behavior of the function near the points where the denominator is zero. For instance, the factors (x + 2) and (x - 1) in the denominator suggest that the function might have vertical asymptotes at x = -2 and x = 1. However, we must always remember the next crucial step: simplification.
Simplifying the Rational Function
Simplifying a rational function is a critical step in the process of identifying vertical asymptotes. After factoring both the numerator and the denominator, we must look for common factors that can be canceled out. This simplification is essential because canceling common factors can reveal the true nature of the function's discontinuities. For the function f(x) = (5x + 5) / (x^2 + x - 2), we factored the numerator as 5(x + 1) and the denominator as (x + 2)(x - 1). So, the function in factored form is f(x) = 5(x + 1) / ((x + 2)(x - 1)). Now, we carefully examine the numerator and the denominator to see if there are any common factors that can be canceled. In this particular case, we observe that there are no common factors between the numerator and the denominator. The factor (x + 1) in the numerator is not present in the denominator, and the factors (x + 2) and (x - 1) in the denominator are not present in the numerator. This means that the function is already in its simplest form. The significance of simplification cannot be overstated. If we had a common factor, such as (x + 1) in both the numerator and the denominator, canceling it would result in a "hole" in the graph of the function at x = -1 rather than a vertical asymptote. A hole is a point of discontinuity where the function is undefined, but the limit of the function exists at that point. On the other hand, a vertical asymptote represents a point where the function approaches infinity. Since our function f(x) = 5(x + 1) / ((x + 2)(x - 1)) is already simplified, we can confidently proceed to identify the vertical asymptotes by finding the zeros of the denominator. This step underscores the importance of meticulous simplification to avoid misinterpreting the discontinuities of a rational function.
Identifying Vertical Asymptotes from the Simplified Denominator
Once the rational function is simplified, the final step in determining the vertical asymptotes involves identifying the values of x that make the denominator equal to zero. These values represent the locations where the function approaches infinity, indicating the presence of vertical asymptotes. For our function, f(x) = 5(x + 1) / ((x + 2)(x - 1)), we have already established that it is in its simplest form. This means that we can directly proceed to find the zeros of the denominator, which is (x + 2)(x - 1). To find the zeros, we set the denominator equal to zero and solve for x: (x + 2)(x - 1) = 0 This equation is satisfied when either (x + 2) = 0 or (x - 1) = 0. Solving these two equations gives us: x + 2 = 0 => x = -2 x - 1 = 0 => x = 1 Therefore, the vertical asymptotes of the function f(x) = 5(x + 1) / ((x + 2)(x - 1)) are x = -2 and x = 1. These values represent the vertical lines that the graph of the function will approach but never touch. At x = -2, the function will either increase towards positive infinity or decrease towards negative infinity as x approaches -2 from the left or the right. Similarly, at x = 1, the function will exhibit similar behavior, either shooting up or down towards infinity. Identifying these vertical asymptotes is crucial for sketching the graph of the function and understanding its behavior. They provide a framework for the graph, guiding its shape and indicating the regions where the function undergoes rapid changes. In summary, by setting the simplified denominator equal to zero and solving for x, we have successfully identified the vertical asymptotes of the function. This final step solidifies our understanding of how rational functions behave near their points of discontinuity.
Conclusion
In conclusion, the vertical asymptotes of the function f(x) = (5x + 5) / (x^2 + x - 2) are x = 1 and x = -2. This determination was achieved through a systematic process involving factoring the numerator and the denominator, simplifying the rational function, and identifying the zeros of the simplified denominator. Vertical asymptotes are essential features of rational functions, indicating where the function becomes unbounded and approaches infinity. Understanding how to find them is crucial for analyzing and graphing rational functions effectively. The steps we followed in this article provide a clear and concise method for identifying vertical asymptotes for any rational function. By mastering these steps, you can confidently tackle similar problems and gain a deeper understanding of the behavior of rational functions. This knowledge is not only valuable in mathematics but also in various fields where rational functions are used to model real-world phenomena. From physics and engineering to economics and finance, the ability to analyze rational functions and identify their asymptotes is a powerful tool. We encourage you to practice identifying vertical asymptotes in different rational functions to solidify your understanding and build your problem-solving skills. Remember, the key is to factor, simplify, and then find the zeros of the denominator. With practice, you will become proficient in this process and develop a strong foundation for further exploration of rational functions and their applications.
