Determining Vertical Asymptotes For Rational Functions G(x) = ((x+5)(x-5)(x-3))/((x+5)(x+3))

Understanding Vertical Asymptotes: In the realm of mathematical functions, vertical asymptotes hold a significant role in defining the behavior of rational functions. A vertical asymptote is essentially an invisible vertical line that a function approaches but never quite touches. These asymptotes are crucial in understanding the function's domain, range, and overall graphical representation. In this comprehensive guide, we will delve deep into the concept of vertical asymptotes, particularly in the context of rational functions, and explore how to identify them. We will use the given function, $g(x)=\frac{(x+5)(x-5)(x-3)}{(x+5)(x+3)}$, as a practical example to illustrate the process of finding vertical asymptotes. Furthermore, we will discuss the importance of simplifying the function before identifying these asymptotes and the common pitfalls to avoid. This knowledge is fundamental for anyone studying calculus, pre-calculus, or any field that involves mathematical modeling and analysis. Mastering the identification of vertical asymptotes not only enhances one's mathematical skills but also provides a deeper understanding of how functions behave near specific points, which is invaluable in various scientific and engineering applications.

Identifying Potential Vertical Asymptotes

To pinpoint the values of $x$ where the function $g(x)$ might exhibit vertical asymptotes, our primary focus lies on the denominator of the rational function. Vertical asymptotes typically occur where the denominator equals zero, as division by zero is undefined in mathematics. This critical step allows us to narrow down the potential locations of these asymptotes. Let's consider the function at hand: $g(x) = \frac{(x+5)(x-5)(x-3)}{(x+5)(x+3)}$. The denominator is $(x+5)(x+3)$. By setting this denominator equal to zero, we create the equation $(x+5)(x+3) = 0$. Solving this equation involves finding the values of $x$ that make each factor zero. Thus, we have two potential solutions: $x+5=0$ and $x+3=0$. Solving these simple equations gives us $x=-5$ and $x=-3$. These values are our candidates for vertical asymptotes. However, it is crucial to remember that these are only potential asymptotes. Further investigation is required to confirm whether they indeed represent vertical asymptotes or if they are removable discontinuities, also known as holes. The next step in our process involves simplifying the function, which will help us differentiate between true vertical asymptotes and removable discontinuities. This step is vital to ensure an accurate understanding of the function's behavior.

Simplifying the Function

Simplifying the function is a critical step in accurately identifying vertical asymptotes. Before definitively declaring a value as a vertical asymptote, we must ensure that the corresponding factor in the denominator does not also appear in the numerator. If a factor exists in both the numerator and the denominator, it can be canceled out, resulting in a 'hole' or a removable discontinuity rather than a vertical asymptote at that x-value. In our example, the function is given by $g(x) = \frac{(x+5)(x-5)(x-3)}{(x+5)(x+3)}$. We observe that the factor $(x+5)$ appears in both the numerator and the denominator. This observation is crucial because it indicates a potential simplification. By canceling out the common factor $(x+5)$, we reduce the function to a simpler form: $g(x) = \frac{(x-5)(x-3)}{(x+3)}$, provided that $x \neq -5$. This simplification is essential because it changes the behavior of the function at $x = -5$. Originally, $x = -5$ made both the numerator and denominator zero, which could have been misinterpreted as a vertical asymptote. However, after simplification, we see that the function is no longer undefined at $x = -5$, but rather has a removable discontinuity or a hole. This step highlights the importance of simplification in accurately determining the vertical asymptotes of a rational function.

Determining the True Vertical Asymptote

After simplifying the function, we are now in a position to accurately determine the vertical asymptotes. As we've seen, the simplified form of our function is $g(x) = \frac{(x-5)(x-3)}{(x+3)}$, with the condition that $x \neq -5$. From our initial analysis of the denominator, we identified $x = -5$ and $x = -3$ as potential vertical asymptotes. However, the simplification process revealed that the factor $(x+5)$ was present in both the numerator and the denominator, leading to a removable discontinuity or a hole at $x = -5$. This means that the function does not have a vertical asymptote at $x = -5$. Now, let's consider the remaining potential asymptote, $x = -3$. In the simplified function, the factor $(x+3)$ remains in the denominator and does not cancel out with any factor in the numerator. This indicates that the function is undefined at $x = -3$, and as $x$ approaches $-3$, the value of the function will approach infinity (or negative infinity). This behavior is characteristic of a vertical asymptote. Therefore, we can definitively conclude that there is a vertical asymptote at $x = -3$. This step-by-step analysis demonstrates the importance of not only identifying potential asymptotes but also verifying them through simplification to avoid misinterpretations.

Conclusion

In summary, to find the vertical asymptotes of the function $g(x) = \frac{(x+5)(x-5)(x-3)}{(x+5)(x+3)}$, we followed a systematic approach. First, we identified potential asymptotes by setting the denominator equal to zero, which gave us $x = -5$ and $x = -3$. Next, we simplified the function by canceling out the common factor $(x+5)$, resulting in $g(x) = \frac{(x-5)(x-3)}{(x+3)}$. This simplification revealed that $x = -5$ is a removable discontinuity (a hole) rather than a vertical asymptote. Finally, we analyzed the simplified function and confirmed that there is a vertical asymptote at $x = -3$ because the factor $(x+3)$ remains in the denominator. Thus, the correct answer to the question "At what value of $x$ will the function $g(x)$ have a vertical asymptote?" is D. -3. This process underscores the crucial steps of identifying potential asymptotes, simplifying the function, and verifying the asymptotes to ensure accuracy. Understanding these steps is essential for mastering the analysis of rational functions and their graphical behavior. The concepts discussed here are fundamental not only in mathematics but also in various fields that utilize mathematical modeling.

Therefore, the answer is D. -3