Finding Discontinuities In Rational Functions Step-by-Step Solution

In the realm of mathematics, rational functions, which are essentially fractions with polynomials in the numerator and denominator, play a crucial role. Understanding the behavior of these functions is paramount, and one of the key aspects to investigate is their continuity. A rational function is continuous everywhere except where the denominator equals zero, as division by zero is undefined. These points where the denominator becomes zero are known as discontinuities. Discontinuities can manifest in different ways, such as vertical asymptotes or holes, each with its own implications for the function's graph and behavior. Identifying these points of discontinuity is essential for a complete analysis of the rational function. This process involves finding the values of 'x' that make the denominator zero, which is typically achieved by factoring the denominator and setting each factor equal to zero. Once these values are found, we can further classify the type of discontinuity by examining the numerator at those points. If the numerator is also zero at the same point, it indicates a hole; otherwise, it represents a vertical asymptote. Understanding these concepts is crucial for various applications, including graphing rational functions, solving equations involving rational expressions, and modeling real-world phenomena. In this comprehensive guide, we will delve deeper into the methods for finding and classifying discontinuities in rational functions, providing you with the tools necessary to master this fundamental mathematical concept. By understanding discontinuities, you gain a deeper insight into the behavior of rational functions and their role in various mathematical and real-world applications. The ability to identify and classify these points is a critical skill for anyone studying calculus, algebra, or related fields. Furthermore, discontinuities play a vital role in understanding the limitations and behaviors of mathematical models, allowing for more accurate predictions and interpretations of results. With this understanding, you can confidently tackle more complex problems involving rational functions and their applications.

To find the points of discontinuity for a rational function, the primary focus is on the denominator. A rational function, expressed as a fraction with polynomials, exhibits discontinuities where the denominator equals zero. This is because division by zero is undefined in mathematics. The process begins by setting the denominator of the rational function equal to zero and then solving for the variable, typically 'x'. This involves algebraic techniques such as factoring, using the quadratic formula, or other methods appropriate for the specific polynomial in the denominator. Factoring is often the most straightforward approach, as it allows us to identify the roots of the polynomial directly. For instance, if the denominator is a quadratic expression, we can try to factor it into two linear factors. Once the denominator is factored, we set each factor equal to zero and solve for 'x'. These values of 'x' are the points of discontinuity. However, it's crucial to remember that not all discontinuities are the same. They can be classified as either vertical asymptotes or holes, depending on the behavior of the numerator at those points. If the numerator is non-zero at a point where the denominator is zero, then there is a vertical asymptote at that point. This means the function approaches infinity (or negative infinity) as 'x' approaches that value. On the other hand, if both the numerator and denominator are zero at the same point, it indicates a hole. A hole is a removable discontinuity, meaning that the function is undefined at that point, but the limit of the function exists as 'x' approaches that value. Understanding the distinction between vertical asymptotes and holes is crucial for accurately graphing and analyzing rational functions. This involves examining the simplified form of the rational function after any common factors in the numerator and denominator have been canceled. The points where the remaining denominator factors are zero correspond to vertical asymptotes, while the canceled factors indicate holes. By systematically analyzing the denominator and considering the behavior of the numerator, we can accurately identify and classify the points of discontinuity in rational functions.

Let's apply the concept of finding points of discontinuity to the given rational function:

$$\frac{x+2}{x^2-16 x+63}$$

The first step is to identify the denominator, which in this case is $x^2 - 16x + 63$. To find the points of discontinuity, we need to determine the values of 'x' that make the denominator equal to zero. This involves solving the quadratic equation: $x^2 - 16x + 63 = 0$. The most common method for solving quadratic equations is factoring. We need to find two numbers that multiply to 63 and add up to -16. These numbers are -7 and -9, so we can factor the quadratic expression as follows:

$$(x - 7)(x - 9) = 0$$

Now, we set each factor equal to zero and solve for 'x':

$$x - 7 = 0 \Rightarrow x = 7$$

$$x - 9 = 0 \Rightarrow x = 9$$

These values, x = 7 and x = 9, are the points where the denominator is zero. Therefore, these are the points of discontinuity for the given rational function. To determine the type of discontinuity, we need to examine the numerator at these points. The numerator is $x + 2$. At x = 7, the numerator is 7 + 2 = 9, which is non-zero. At x = 9, the numerator is 9 + 2 = 11, which is also non-zero. Since the numerator is non-zero at both x = 7 and x = 9, these points correspond to vertical asymptotes. This means that the function will approach infinity (or negative infinity) as 'x' approaches 7 and 9. There are no holes in this rational function because there are no common factors between the numerator and the denominator. The points of discontinuity are solely determined by the zeros of the denominator, and in this case, they both result in vertical asymptotes. Therefore, the points of discontinuity for the given rational function are x = 7 and x = 9. This analysis provides a complete understanding of the function's behavior around these points, which is essential for graphing and further analysis of the function.

Now, let's analyze the given options based on our findings:

• A. x = -2: This option is incorrect because x = -2 is a zero of the numerator, not the denominator. The points of discontinuity are determined by the zeros of the denominator.
• B. x = -7, x = -9: This option is incorrect because these values do not make the denominator zero. The correct values are x = 7 and x = 9.
• C. x = 7, x = 9: This option is correct. As we calculated, the denominator is zero when x = 7 and x = 9, making these the points of discontinuity.
• D. x = 7, x = -9: This option is incorrect because while x = 7 is a point of discontinuity, x = -9 is not. The correct pair is x = 7 and x = 9.

Therefore, the correct answer is C. x = 7, x = 9. This option accurately identifies the points where the denominator of the rational function is zero, which are the points of discontinuity. Understanding why the other options are incorrect is just as important as identifying the correct answer. Option A is a common mistake, confusing the zeros of the numerator with the zeros of the denominator. Options B and D demonstrate a misunderstanding of the factoring process or an error in solving the quadratic equation. By carefully analyzing the function and applying the correct algebraic techniques, we can confidently determine the points of discontinuity and avoid these common pitfalls. This methodical approach ensures accuracy and a deeper understanding of the function's behavior. In summary, the process of analyzing the options reinforces the importance of a systematic approach to solving mathematical problems. Each option provides an opportunity to review the concepts and techniques involved, leading to a more robust understanding of the material.

In conclusion, finding the points of discontinuity for a rational function is a crucial skill in mathematical analysis. These points, where the function is undefined due to division by zero, provide valuable insights into the function's behavior and characteristics. The process involves identifying the denominator, setting it equal to zero, and solving for the variable. This typically involves algebraic techniques such as factoring, using the quadratic formula, or other appropriate methods. Once the points of discontinuity are identified, it's essential to classify them as either vertical asymptotes or holes. Vertical asymptotes occur when the denominator is zero, and the numerator is non-zero, leading to the function approaching infinity (or negative infinity). Holes, on the other hand, occur when both the numerator and denominator are zero, representing a removable discontinuity. Understanding the distinction between these types of discontinuities is crucial for accurately graphing and analyzing rational functions. The given example, $\frac{x+2}{x^2-16 x+63}$, illustrates this process effectively. By factoring the denominator, we found the points of discontinuity to be x = 7 and x = 9. Since the numerator was non-zero at these points, we classified them as vertical asymptotes. This step-by-step approach ensures accuracy and a thorough understanding of the function's behavior. Furthermore, this skill is not limited to theoretical mathematics; it has practical applications in various fields, including physics, engineering, and economics, where rational functions are used to model real-world phenomena. The ability to identify and interpret discontinuities is essential for making accurate predictions and informed decisions based on these models. Therefore, mastering the concept of discontinuities in rational functions is a valuable investment in your mathematical toolkit, providing you with the tools and understanding necessary to tackle a wide range of problems and applications.