\( \frac{4 x}{x-3} \) and \( \frac{x}{x-5} \)

The expression \( \frac{4 x}{x-3} \) represents a rational function, which can be visualized as a hyperbola with vertical and horizontal asymptotes. The vertical asymptote occurs when the denominator is zero, specifically at \( x = 3 \), limiting the function's values as the input approaches this point. Similarly, \( \frac{x}{x-5} \) is another rational function with a vertical asymptote at \( x = 5 \). Understanding these asymptotes helps predict the function's behavior when graphed. In practical scenarios, these types of functions can often model real-life relationships, such as speeds where a numerator represents distance traveled and a denominator represents time. For instance, if you're analyzing how speed varies as obstacles (represented by distances in the denominator) get closer, you can observe behaviors that might help inform decisions, such as optimizing routes to avoid delays.