Vertical Asymptotes Of G(x) = 1/(x-5)^2 A Step-by-Step Guide

In the realm of mathematical functions, vertical asymptotes play a crucial role in understanding the behavior and graphical representation of rational functions. A vertical asymptote is a vertical line that a function approaches but never quite touches, indicating a point where the function's value tends towards infinity or negative infinity. Identifying vertical asymptotes is essential for sketching accurate graphs and analyzing the function's domain and range. In this article, we will delve into the process of finding the vertical asymptotes of the function $g(x) = \frac{1}{(x-5)^2}$, providing a step-by-step guide and elucidating the underlying concepts.

Understanding Vertical Asymptotes

Before we embark on the specific example, let's establish a firm understanding of what vertical asymptotes are and why they occur. Vertical asymptotes typically arise in rational functions, which are functions expressed as the ratio of two polynomials, such as $g(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. A vertical asymptote occurs at any value of $x$ for which the denominator $Q(x)$ equals zero, while the numerator $P(x)$ does not equal zero. This is because division by zero is undefined, causing the function's value to approach infinity or negative infinity as $x$ approaches that particular value.

Graphically, a vertical asymptote is represented by a vertical dashed line. As the graph of the function gets closer and closer to this line, it either shoots up towards positive infinity or plummets down towards negative infinity. The function never actually crosses the vertical asymptote, but it gets arbitrarily close. The vertical asymptotes essentially mark the boundaries of the function's domain, indicating the values of $x$ that are not permissible.

Step 1: Identify the Denominator

The first step in finding the vertical asymptotes of a rational function is to identify the denominator. In the case of our function, $g(x) = \frac{1}{(x-5)^2}$, the denominator is $(x-5)^2$. The denominator is the key to unlocking the location of vertical asymptotes, as it's the part of the function that can potentially cause division by zero.

The denominator in a rational function plays a critical role in determining the function's behavior, especially in the context of vertical asymptotes. A rational function is essentially a fraction where both the numerator and the denominator are polynomials. The denominator, being the divisor in this fraction, dictates the values of $x$ for which the function is defined. When the denominator equals zero, the function becomes undefined, leading to the potential formation of a vertical asymptote. It's important to remember that the numerator also plays a role; if both the numerator and denominator are zero at the same point, it might indicate a hole in the graph rather than a vertical asymptote. However, in most cases, finding the zeros of the denominator is the primary step in identifying vertical asymptotes.

Step 2: Set the Denominator Equal to Zero

Once we have identified the denominator, the next step is to set it equal to zero. This is because the vertical asymptotes occur at the values of $x$ that make the denominator zero. For our function, we set $(x-5)^2 = 0$. This equation represents the condition that leads to the function becoming undefined, and solving it will reveal the x-values where the vertical asymptotes are located. This step is crucial because it directly links the algebraic representation of the function to its graphical behavior, specifically highlighting where the function will exhibit unbounded growth or decay. By setting the denominator to zero, we are essentially finding the roots or zeros of the denominator polynomial, which correspond to the x-coordinates of the vertical asymptotes.

Step 3: Solve for x

Now, we need to solve the equation $(x-5)^2 = 0$ for $x$. This equation is a simple quadratic equation, and we can solve it by taking the square root of both sides. Taking the square root of both sides, we get $x-5 = 0$. Adding 5 to both sides, we find $x = 5$. This is the value of $x$ where the denominator is zero, and hence, it's a potential location for a vertical asymptote. Solving for $x$ in this step is the core of finding the vertical asymptotes; it translates the algebraic condition of the denominator being zero into a specific numerical value on the x-axis. This value represents the line $x = 5$, which the function will approach but never touch, illustrating the fundamental characteristic of a vertical asymptote.

Step 4: Check for Non-Removable Discontinuities

Before declaring $x = 5$ as a vertical asymptote, we need to ensure that it's not a removable discontinuity, also known as a hole. A removable discontinuity occurs when both the numerator and denominator of the rational function are zero at the same value of $x$. To check for this, we would evaluate the numerator at $x = 5$. In our case, the numerator is 1, which is not zero. Therefore, $x = 5$ is indeed a vertical asymptote.

Checking for non-removable discontinuities is a crucial step in accurately identifying vertical asymptotes. It involves ensuring that the factor that makes the denominator zero does not also make the numerator zero. If both the numerator and the denominator share a common factor that becomes zero at a particular $x$ value, it results in a hole in the graph rather than a vertical asymptote. This is because the common factor can be cancelled out, effectively