Finding Vertical Asymptotes Of F(x) = (x-9)/(x^3-81x)

In the realm of mathematical functions, especially within the fascinating domain of rational functions, vertical asymptotes hold a position of paramount importance. These invisible lines, acting as guiding boundaries, offer profound insights into the behavior of functions as their input values approach specific points. A vertical asymptote emerges at a particular x-value where the function's output surges towards positive or negative infinity. Unveiling these asymptotes involves a meticulous examination of the function's structure, particularly its denominator, seeking those critical points where it approaches zero.

To embark on this investigative journey, let's consider the function f(x) = (x-9)/(x³-81x). Our mission is to navigate through its intricate landscape and pinpoint its vertical asymptotes. This exploration will not only illuminate the nature of this specific function but also deepen our understanding of how vertical asymptotes manifest in rational functions in general. We'll dissect the function, scrutinize its components, and employ algebraic techniques to unveil the hidden lines that dictate its behavior.

Our initial step in identifying vertical asymptotes lies in the meticulous analysis of the function f(x) = (x-9)/(x³-81x). This involves a strategic maneuver: factoring both the numerator and the denominator. Factoring serves as a powerful lens, allowing us to discern common factors and, more importantly, to expose the roots of the denominator. These roots, the values of x that render the denominator zero, are the prime suspects in our quest for vertical asymptotes.

The numerator, (x-9), presents itself in its simplest form, a linear expression ready for action. However, the denominator, (x³-81x), beckons for factorization. We embark on this algebraic endeavor by first extracting the common factor x, transforming the denominator into x(x²-81). The expression within the parentheses, (x²-81), reveals itself as a difference of squares, a classic algebraic pattern that can be further factored into (x-9)(x+9). Thus, the fully factored denominator takes the form x(x-9)(x+9). Now, our function f(x) stands revealed in its factored glory: (x-9) / [x(x-9)(x+9)].

With the function now elegantly factored as f(x) = (x-9) / [x(x-9)(x+9)], we are poised to identify the potential vertical asymptotes. These elusive lines reside at the x-values that make the denominator vanish into zero. By setting each factor in the denominator to zero, we embark on a quest to unearth these critical points.

The equation x = 0 immediately reveals one such point, a potential vertical asymptote lurking at the origin. Next, we consider the factor (x-9). Setting this to zero yields x = 9, another candidate for a vertical asymptote. Finally, the factor (x+9) leads us to x = -9, completing our trio of potential asymptotes. At this stage, we have identified three contenders: x = 0, x = 9, and x = -9. However, the journey is not yet complete. We must exercise caution and investigate further, as the presence of a factor in both the numerator and denominator can lead to a removable discontinuity, a "hole" in the graph rather than a true vertical asymptote.

Before we definitively declare our vertical asymptotes, we must engage in the crucial step of simplifying the function. Simplification involves identifying and canceling out any common factors that grace both the numerator and the denominator. These shared factors, while initially appearing to contribute to vertical asymptotes, often reveal themselves as harbingers of removable discontinuities, those intriguing "holes" in the graph of the function.

In our function, f(x) = (x-9) / [x(x-9)(x+9)], the factor (x-9) makes a conspicuous appearance in both the numerator and the denominator. This shared factor signals a potential removable discontinuity at x = 9. By canceling out this common factor, we transform our function into its simplified form: f(x) = 1 / [x(x+9)]. This simplification unveils the true nature of the function, stripping away the disguise of the removable discontinuity and revealing the genuine vertical asymptotes.

With the function now streamlined to its simplest form, f(x) = 1 / [x(x+9)], we stand ready to definitively declare the vertical asymptotes. These are the vertical lines that the function approaches infinitely closely but never quite touches, the boundaries that dictate its behavior as x approaches specific values.

In our simplified function, the denominator x(x+9) holds the key to unlocking these asymptotes. Setting each factor to zero, we find the values of x that cause the denominator to vanish. The equation x = 0 reveals a vertical asymptote at the y-axis. The factor (x+9), when set to zero, yields x = -9, unveiling another vertical asymptote lurking at this location.

The value x = 9, which initially appeared as a potential vertical asymptote, has been revealed as a removable discontinuity due to the cancellation of the (x-9) factor. This means that at x = 9, the function has a "hole," a point where it is undefined, rather than an asymptote.

Therefore, the vertical asymptotes of the function f(x) = (x-9) / (x³-81x) are definitively x = 0 and x = -9. These lines act as guiding rails, shaping the graph of the function and dictating its behavior as x approaches these critical values. The function will surge towards positive or negative infinity as it nears these asymptotes, never quite crossing the boundary.

Our exploration of the function f(x) = (x-9) / (x³-81x) has been a journey into the heart of rational functions and their fascinating features, particularly vertical asymptotes. We embarked on this quest by meticulously factoring the function, a crucial step in unveiling the roots of the denominator. These roots, the values of x that make the denominator zero, served as our initial suspects in the hunt for vertical asymptotes.

However, the journey was not without its twists and turns. We encountered the concept of removable discontinuities, those "holes" in the graph that can masquerade as asymptotes. Through simplification, we learned to distinguish true vertical asymptotes from these imposters, ensuring the accuracy of our findings.

Ultimately, we declared the vertical asymptotes of f(x) to be x = 0 and x = -9. These lines, invisible yet powerful, shape the function's graph and dictate its behavior as x approaches these critical values. Our exploration serves as a testament to the intricate beauty of rational functions and the profound insights that vertical asymptotes offer into their nature. By mastering the techniques of factoring, simplification, and root identification, we can confidently navigate the world of rational functions and unravel their hidden structures.

Therefore, the final answer is A. x=0,-9.