a) \( f(x)=\frac{x}{2-x} \)

To get a good grasp of the function \( f(x) = \frac{x}{2-x} \), it's fun to analyze its behavior across different values of \( x \). Notice that as \( x \) approaches 2, the denominator gets closer to zero, causing the function to head towards infinity or negative infinity (depending on the direction of approach). This point is a vertical asymptote! Fun fact: vertical asymptotes are like the bouncers of the graph, keeping the function away from certain parts of the coordinate plane. Now, this function is also a great candidate for real-world scenarios involving ratios, such as modeling proportions in resource allocation. For instance, if you’re distributing resources where the total availability is fixed (like how much food is given out in a community), \( f(x) \) can help you understand the impact of changing one variable—like how increasing the amount of resources available will change the distribution ratios. It's like having a secret tool to optimize your decision-making!