Identifying Holes And Vertical Asymptotes Of F(x) = (x+99)(x+95) / (x+10)(x+95)(x+60)

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In the realm of mathematical functions, rational functions hold a special place, often presenting intriguing behaviors and characteristics. Among these, discontinuities – points where the function's graph breaks or behaves erratically – are particularly noteworthy. These discontinuities manifest as either holes (removable discontinuities) or vertical asymptotes (non-removable discontinuities), each carrying distinct implications for the function's behavior.

This article delves into the fascinating world of discontinuities in rational functions, specifically focusing on the function f(x) = (x+99)(x+95) / (x+10)(x+95)(x+60). We will embark on a journey to identify the values of x for which this function exhibits holes and vertical asymptotes, providing a comprehensive understanding of its behavior around these critical points.

Deciphering Discontinuities: Holes vs. Vertical Asymptotes

Before we plunge into the specifics of our function, let's first establish a firm grasp on the fundamental concepts of holes and vertical asymptotes. These two types of discontinuities arise from different scenarios and have distinct visual representations on the graph of a rational function.

Holes: The Removable Discontinuities

Holes, also known as removable discontinuities, occur when a factor in the numerator and denominator of a rational function cancels out. This cancellation creates a "hole" in the graph at the x-value that makes the canceled factor equal to zero. In essence, the function is undefined at this specific point, but the limit of the function as x approaches this value exists.

To visualize a hole, imagine a smooth curve with a tiny gap or puncture at a particular point. The function behaves normally everywhere else, but at this specific x-value, it's as if a small piece of the graph has been removed. The x-coordinate of the hole is the value that makes the canceled factor zero, and the y-coordinate can be found by evaluating the simplified function (after cancellation) at that x-value.

Vertical Asymptotes: The Non-Removable Barriers

Vertical asymptotes, on the other hand, are non-removable discontinuities that occur when the denominator of a rational function equals zero, but the numerator does not. At these points, the function's value approaches infinity (or negative infinity), causing the graph to shoot up or down vertically, never actually crossing the vertical asymptote line.

Think of a vertical asymptote as an invisible barrier that the function's graph cannot cross. As x gets closer and closer to the x-value of the asymptote, the function's value grows without bound, either positively or negatively. The graph will get arbitrarily close to the asymptote line but never touch or intersect it.

Unraveling the Discontinuities of f(x) = (x+99)(x+95) / (x+10)(x+95)(x+60)

Now that we have a clear understanding of holes and vertical asymptotes, let's apply these concepts to our function: f(x) = (x+99)(x+95) / (x+10)(x+95)(x+60). Our goal is to pinpoint the x-values where this function exhibits these discontinuities.

Identifying Potential Discontinuities

The first step in identifying discontinuities is to examine the denominator of the rational function. The denominator, (x+10)(x+95)(x+60), reveals potential points of discontinuity where it equals zero. Setting each factor equal to zero, we find:

  • x + 10 = 0 => x = -10
  • x + 95 = 0 => x = -95
  • x + 60 = 0 => x = -60

These three values, x = -10, x = -95, and x = -60, are potential locations for discontinuities. However, we need to further investigate whether these discontinuities are holes or vertical asymptotes.

Distinguishing Holes from Vertical Asymptotes

To determine the nature of each discontinuity, we must analyze whether the corresponding factor in the denominator also appears in the numerator. If a factor cancels out, it indicates a hole; if it remains in the denominator, it indicates a vertical asymptote.

In our function, the factor (x+95) appears in both the numerator and the denominator. This means that it cancels out, creating a hole at x = -95. To find the y-coordinate of the hole, we substitute x = -95 into the simplified function after cancellation:

Simplified function: f(x) = (x+99) / (x+10)(x+60)

f(-95) = (-95 + 99) / (-95 + 10)(-95 + 60) = 4 / (-85)(-35) = 4 / 2975

Therefore, there is a hole at the point (-95, 4/2975).

The remaining factors in the denominator, (x+10) and (x+60), do not appear in the numerator. This indicates that they correspond to vertical asymptotes. Thus, there are vertical asymptotes at x = -10 and x = -60.

Summarizing the Discontinuities

In summary, the function f(x) = (x+99)(x+95) / (x+10)(x+95)(x+60) exhibits the following discontinuities:

  • Hole: at x = -95
  • Vertical Asymptotes: at x = -10 and x = -60

Visualizing the Discontinuities: Graphing the Function

To solidify our understanding, let's consider the graphical representation of the function. The graph will visually demonstrate the presence of the hole and the vertical asymptotes.

  • Hole: At x = -95, there will be a small gap or break in the graph. The function will approach this point from both sides but will not be defined at x = -95.
  • Vertical Asymptotes: At x = -10 and x = -60, the graph will approach vertical lines. As x gets closer to these values, the function's value will shoot up or down towards infinity (or negative infinity), never crossing the lines x = -10 and x = -60.

By visualizing the graph, we can see how the algebraic analysis of the function translates into its graphical behavior. The hole represents a removable discontinuity, while the vertical asymptotes represent non-removable barriers.

Significance of Discontinuities in Mathematical Analysis

The identification and analysis of discontinuities are crucial in various areas of mathematics, particularly in calculus and real analysis. Discontinuities can significantly impact the behavior of a function, affecting its differentiability, integrability, and overall properties.

For instance, the presence of vertical asymptotes can lead to improper integrals, where the integral's value may be infinite. Holes, while removable, can still affect the function's behavior in specific contexts, such as when evaluating limits or defining continuity.

Understanding discontinuities allows us to gain a deeper insight into the function's nature and predict its behavior in different scenarios. This knowledge is essential for solving mathematical problems, modeling real-world phenomena, and making informed decisions based on mathematical analysis.

Conclusion: Mastering the Art of Discontinuity Detection

In this article, we have embarked on a journey to unravel the discontinuities of the rational function f(x) = (x+99)(x+95) / (x+10)(x+95)(x+60). We have learned to distinguish between holes and vertical asymptotes, identifying them through algebraic analysis and visualizing them graphically.

The key takeaway is that holes arise from factors that cancel out in the numerator and denominator, while vertical asymptotes stem from factors that remain in the denominator. By carefully examining the function's structure, we can confidently pinpoint these discontinuities and understand their implications for the function's behavior.

Mastering the art of discontinuity detection is a valuable skill in mathematical analysis. It empowers us to delve deeper into the nature of functions, predict their behavior, and apply this knowledge to solve complex problems in various fields.

Answering the Question: Holes and Vertical Asymptotes of f(x)

To directly answer the question posed, the function f(x) = (x+99)(x+95) / (x+10)(x+95)(x+60) has:

  • A hole at x = -95
  • Vertical asymptotes at x = -10 and x = -60

This comprehensive analysis provides a clear understanding of the function's discontinuities, both algebraically and graphically.