X-Intercepts Of Rational Function R(x) = (x+1)(x-2) / (x+2)(x-9)

Rational functions, a cornerstone of algebraic functions, take the form of a ratio between two polynomials. Delving into their characteristics, especially x-intercepts, unveils crucial insights into their behavior and graphical representation. In this article, we will focus on the rational function r(x) = (x+1)(x-2) / (x+2)(x-9), meticulously determining its x-intercepts and unraveling their significance. An x-intercept, also known as a root or zero of a function, signifies the point where the graph intersects the x-axis. At these points, the function's value, r(x), equals zero. For rational functions, this condition occurs precisely when the numerator equals zero, provided the denominator does not simultaneously equal zero. Understanding x-intercepts is essential for sketching the graph of a rational function, analyzing its domain and range, and identifying key features such as vertical asymptotes and holes. These intercepts provide the foundation for understanding where the function's output is zero, marking crucial points of transition in the function's behavior. To determine the x-intercepts of r(x), we will set the numerator, (x+1)(x-2), to zero and solve for x. This process will reveal the specific x-values at which the function crosses the x-axis, giving us a clear picture of its behavior around those points. Furthermore, examining the denominator (x+2)(x-9) allows us to identify any potential restrictions on the domain, such as vertical asymptotes, which are also crucial for understanding the overall behavior of the rational function. The interplay between the numerator and denominator determines the function's x-intercepts and vertical asymptotes, shaping its unique characteristics and graphical representation. By carefully analyzing the function r(x), we can gain a comprehensive understanding of how these elements contribute to its behavior and provide valuable insights into the broader class of rational functions. This exploration will not only enhance our understanding of this specific function but also equip us with the tools to analyze other rational functions effectively.

Finding the X-Intercepts of r(x) = (x+1)(x-2) / (x+2)(x-9)

To pinpoint the x-intercepts of the rational function r(x) = (x+1)(x-2) / (x+2)(x-9), we focus on the numerator. The x-intercepts occur when the function's value equals zero, which, in a rational function, happens when the numerator is zero (provided the denominator is not simultaneously zero). Therefore, we need to solve the equation (x+1)(x-2) = 0. This equation is already factored, making it straightforward to find the solutions. We set each factor equal to zero: x + 1 = 0 and x - 2 = 0. Solving x + 1 = 0, we subtract 1 from both sides to get x = -1. Solving x - 2 = 0, we add 2 to both sides to get x = 2. These two values, x = -1 and x = 2, are the potential x-intercepts of the function. However, we must ensure that these values do not also make the denominator zero, as that would indicate a hole rather than an x-intercept. The denominator is (x+2)(x-9). If x = -1, the denominator becomes (-1+2)(-1-9) = (1)(-10) = -10, which is not zero. If x = 2, the denominator becomes (2+2)(2-9) = (4)(-7) = -28, which is also not zero. Since neither x = -1 nor x = 2 makes the denominator zero, we can confidently conclude that these are indeed the x-intercepts of the function. The x-intercepts are the points where the graph of the function crosses the x-axis. In this case, the graph crosses the x-axis at x = -1 and x = 2. These points provide valuable information about the function's behavior and are essential for sketching its graph. By identifying these intercepts, we gain a deeper understanding of how the function behaves around these crucial points. Furthermore, knowing the x-intercepts helps us to analyze the intervals where the function is positive or negative, which is crucial for a complete understanding of its behavior. This detailed analysis of the numerator and denominator allows us to accurately determine the x-intercepts and avoid any potential pitfalls, such as misidentifying holes as intercepts. Therefore, the x-intercepts of r(x) = (x+1)(x-2) / (x+2)(x-9) are x = -1 and x = 2.

Identifying the Smaller and Larger X-Intercepts

Having determined the x-intercepts of the rational function r(x) = (x+1)(x-2) / (x+2)(x-9) to be x = -1 and x = 2, the next step is to identify which value is smaller and which is larger. This is a straightforward comparison: -1 is less than 2. Therefore, the smaller x-intercept is x = -1, and the larger x-intercept is x = 2. Understanding the relative positions of these intercepts on the number line is crucial for sketching the graph of the function. The smaller x-intercept, x = -1, lies to the left of the larger x-intercept, x = 2, on the x-axis. This positioning helps us to visualize the behavior of the function as it crosses the x-axis at these points. Furthermore, knowing the order of the x-intercepts allows us to analyze the intervals where the function is positive or negative more effectively. For instance, between the two x-intercepts (-1 and 2), the function's value will either be consistently positive or consistently negative. To determine the sign of the function in this interval, we can choose a test value between -1 and 2, such as x = 0, and evaluate r(0). Similarly, we can analyze the function's behavior in the intervals to the left of x = -1 and to the right of x = 2. This information, combined with the knowledge of the x-intercepts and vertical asymptotes, provides a comprehensive understanding of the function's graph. The ability to identify and order the x-intercepts is a fundamental skill in analyzing rational functions. It allows us to break down the function's behavior into manageable intervals and understand how it transitions between positive and negative values. In the context of r(x) = (x+1)(x-2) / (x+2)(x-9), recognizing that x = -1 is the smaller x-intercept and x = 2 is the larger x-intercept is a key step in fully understanding its characteristics and graphical representation. This careful analysis ensures we have a solid foundation for further exploration of the function's properties.

Conclusion: Significance of X-Intercepts in Rational Function Analysis

In conclusion, we have successfully determined the x-intercepts of the rational function r(x) = (x+1)(x-2) / (x+2)(x-9). By setting the numerator equal to zero and solving for x, we found the x-intercepts to be x = -1 and x = 2. We further identified x = -1 as the smaller x-intercept and x = 2 as the larger x-intercept. These x-intercepts are crucial points where the graph of the function intersects the x-axis, providing valuable information about the function's behavior and characteristics. The x-intercepts, also known as roots or zeros of the function, are fundamental in understanding the intervals where the function's value is positive, negative, or zero. They serve as critical reference points for sketching the graph of the function, as they delineate the regions above and below the x-axis. Additionally, x-intercepts are instrumental in solving inequalities involving rational functions. By identifying the x-intercepts and vertical asymptotes, we can create a sign chart that helps determine the intervals where the function satisfies a given inequality. This is particularly useful in various applications, such as optimization problems and analyzing the behavior of systems modeled by rational functions. Moreover, the x-intercepts, in conjunction with the vertical asymptotes, help define the domain of the function. Vertical asymptotes occur where the denominator of the rational function equals zero, and these points are excluded from the domain. The x-intercepts, on the other hand, are included in the domain unless they also make the denominator zero (in which case they represent holes in the graph). Understanding the relationship between the x-intercepts and vertical asymptotes is essential for a comprehensive analysis of the function's domain and range. The process of finding x-intercepts reinforces key algebraic skills, such as factoring and solving equations. It also highlights the importance of careful analysis and attention to detail, as neglecting to check for common factors between the numerator and denominator can lead to misidentification of x-intercepts and holes. Therefore, the ability to accurately determine the x-intercepts of a rational function is a cornerstone of mathematical proficiency and a valuable tool in various fields of study and application.

Final Answer: The function r has x-intercepts -1 (smaller value) and 2 (larger value).