Analyzing F(x) = (x+6) / (4x^2 - 9x + 5) Domain, Intercepts, And Asymptotes

This article delves into a comprehensive analysis of the function f(x) = (x+6) / (4x^2 - 9x + 5). We will explore its domain, intercepts, asymptotes, and other key features. Understanding these elements provides a complete picture of the function's behavior and its graphical representation. Mastering function analysis is crucial for various mathematical and scientific applications, from modeling physical phenomena to optimizing engineering designs. This exploration serves as a valuable resource for students, educators, and anyone interested in gaining a deeper understanding of mathematical functions. By the end of this article, you will have a solid grasp of how to dissect a rational function and identify its critical characteristics. We'll start by defining the function, then systematically examine each aspect, providing clear explanations and examples along the way. This step-by-step approach ensures that even those with a basic mathematical background can follow the analysis and appreciate the elegance and power of function analysis.

H2: 1) Decoding the Domain of f(x)

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the function f(x) = (x+6) / (4x^2 - 9x + 5), we need to identify any values of x that would make the denominator equal to zero, as division by zero is undefined. Therefore, we need to solve the equation 4x^2 - 9x + 5 = 0. Factoring the quadratic expression, we get (x-1)(4x-5) = 0. This gives us two solutions: x = 1 and x = 5/4. These are the values that make the denominator zero, and hence, they must be excluded from the domain. Expressing the domain in interval notation, we have (-∞, 1) ∪ (1, 5/4) ∪ (5/4, ∞). This notation indicates that the domain includes all real numbers except for x = 1 and x = 5/4. Understanding the domain is crucial because it defines the boundaries within which the function operates meaningfully. Outside of this domain, the function produces undefined results. In practical applications, the domain might represent physical constraints or limitations on the inputs to a model. For instance, if x represents time, negative values might not be meaningful, thereby restricting the domain to non-negative numbers. Similarly, in engineering contexts, the domain might be constrained by material properties or system limitations. Therefore, identifying the domain is not just a mathematical exercise; it often has significant implications for the interpretation and application of the function in real-world scenarios. Recognizing the domain helps to avoid nonsensical results and ensures that the function is used within its valid range. This initial step sets the stage for further analysis, as it establishes the foundation upon which the rest of the function's behavior is built. By carefully examining the denominator, we have pinpointed the values that must be excluded, giving us a clear picture of the function's permissible inputs.

H2: 2) Pinpointing the Y-Intercept of f(x)

The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when x = 0. To find the y-intercept of f(x) = (x+6) / (4x^2 - 9x + 5), we substitute x = 0 into the function: f(0) = (0+6) / (4(0)^2 - 9(0) + 5) = 6/5. Therefore, the y-intercept is at the point (0, 6/5). The y-intercept provides a crucial reference point for visualizing the function's graph. It tells us where the function's value is when the input is zero. This is often a significant value in practical contexts. For example, if the function represents the amount of a substance over time, the y-intercept would indicate the initial amount of the substance. Similarly, in economic models, the y-intercept might represent the fixed costs that are incurred even when no production takes place. The y-intercept can also give us insights into the function's overall behavior. If the y-intercept is a large value, it suggests that the function's values are generally high near x = 0. Conversely, a y-intercept close to zero indicates that the function's values are relatively small in this region. Furthermore, the y-intercept is an easy-to-calculate point that can help in sketching a rough graph of the function. By combining the y-intercept with other key features, such as the x-intercepts and asymptotes, we can gain a more complete understanding of the function's graphical representation. In summary, finding the y-intercept is a straightforward but valuable step in function analysis. It provides a critical point on the graph and offers insights into the function's behavior near x = 0. This information is essential for both mathematical understanding and practical applications of the function.

H2: 3) Locating the X-Intercept(s) of f(x)

The x-intercept(s) are the points where the graph of the function intersects the x-axis. These occur when f(x) = 0. For the function f(x) = (x+6) / (4x^2 - 9x + 5), we need to solve the equation (x+6) / (4x^2 - 9x + 5) = 0. A fraction is equal to zero only when its numerator is zero (and the denominator is not zero). Thus, we need to solve x+6 = 0, which gives us x = -6. Now we need to verify that the denominator is not zero at x = -6. Substituting x = -6 into the denominator, we get 4(-6)^2 - 9(-6) + 5 = 4(36) + 54 + 5 = 144 + 54 + 5 = 203, which is not zero. Therefore, the x-intercept is at the point (-6, 0). The x-intercepts are particularly important because they indicate the values of x for which the function's output is zero. In many applications, these values have specific meanings. For instance, if the function represents the profit of a business, the x-intercepts would represent the break-even points where the profit is zero. Similarly, in physics, the x-intercepts might indicate the equilibrium positions of a system. The x-intercepts also provide crucial information for sketching the graph of the function. They help to anchor the graph to the x-axis and show where the function changes sign (from positive to negative or vice versa). Together with the y-intercept and the asymptotes, the x-intercepts provide a framework for understanding the overall shape and behavior of the function. It's important to check that the values obtained from setting the numerator to zero do not also make the denominator zero. If a value makes both the numerator and denominator zero, it might indicate a hole in the graph rather than an x-intercept. In this case, however, x = -6 only makes the numerator zero, so it is a valid x-intercept. In summary, finding the x-intercepts is a key step in function analysis. They represent the points where the function's output is zero and provide valuable insights into the function's behavior and its graphical representation. By identifying these points, we gain a more complete understanding of the function's properties and its potential applications.

H2: 4) Identifying the Vertical Asymptotes of f(x)

Vertical asymptotes occur at x-values where the function approaches infinity or negative infinity. For rational functions, these typically occur where the denominator is zero and the numerator is non-zero. We already found that the denominator of f(x) = (x+6) / (4x^2 - 9x + 5), which is (x-1)(4x-5), is zero at x = 1 and x = 5/4. The numerator, x+6, is not zero at these points. Therefore, the vertical asymptotes are at x = 1 and x = 5/4. Vertical asymptotes are fundamental features of rational functions, as they dictate the function's behavior near certain x-values. A vertical asymptote indicates that the function's value becomes infinitely large (either positively or negatively) as x approaches a specific value. This behavior is crucial for understanding the function's graph and its limitations. In the context of real-world applications, vertical asymptotes can represent physical constraints or boundaries. For example, in a chemical reaction, a vertical asymptote might indicate a concentration limit that cannot be exceeded. Similarly, in economics, it might represent a production capacity limit. Identifying vertical asymptotes involves finding the zeros of the denominator and ensuring that the numerator is not also zero at those points. If both the numerator and denominator are zero, it might indicate a hole in the graph rather than a vertical asymptote. The vertical asymptotes divide the domain of the function into intervals, and the function's behavior can differ significantly across these intervals. Therefore, understanding the vertical asymptotes is essential for sketching an accurate graph of the function and for interpreting its behavior in different regions of the domain. Furthermore, vertical asymptotes are closely related to the concept of limits. The limit of the function as x approaches a vertical asymptote will be either positive infinity, negative infinity, or undefined. This connection highlights the importance of understanding limits in the context of function analysis. In summary, identifying the vertical asymptotes is a critical step in analyzing rational functions. They define the x-values where the function exhibits extreme behavior and provide valuable information for understanding the function's graph and its practical applications. By locating these asymptotes, we gain a deeper insight into the function's characteristics and its limitations.

H3: Conclusion and Summary of Findings

In summary, we have conducted a comprehensive analysis of the function f(x) = (x+6) / (4x^2 - 9x + 5). We determined the domain to be (-∞, 1) ∪ (1, 5/4) ∪ (5/4, ∞), excluding the values x = 1 and x = 5/4 where the denominator is zero. The y-intercept was found to be at the point (0, 6/5), indicating the function's value when x = 0. The x-intercept was identified as (-6, 0), representing the point where the function's graph crosses the x-axis. Finally, we located the vertical asymptotes at x = 1 and x = 5/4, signifying the points where the function approaches infinity. This detailed analysis provides a complete picture of the function's behavior and its key characteristics. Understanding the domain ensures that we only consider valid input values. The intercepts give us specific points on the graph that help to orient its position. The vertical asymptotes highlight the function's behavior near certain x-values, showing where it becomes unbounded. By combining all these elements, we can sketch an accurate graph of the function and interpret its behavior in various contexts. Function analysis is a powerful tool in mathematics and its applications. It allows us to understand complex relationships and predict how functions will behave under different conditions. This skill is essential in fields such as physics, engineering, economics, and computer science, where mathematical models are used to represent real-world phenomena. The process of analyzing a function involves systematically examining its various properties, such as its domain, intercepts, asymptotes, and limits. By breaking down the function into its constituent parts, we can gain a deeper understanding of its overall behavior. This approach is not only useful for understanding specific functions but also for developing a general problem-solving strategy in mathematics. In conclusion, the analysis of f(x) = (x+6) / (4x^2 - 9x + 5) serves as a valuable example of how to dissect a rational function and identify its key features. By mastering these techniques, we can confidently tackle more complex functions and apply our knowledge to a wide range of mathematical and practical problems. This comprehensive understanding empowers us to make informed decisions and predictions based on mathematical models.