Q4-Sketch a graph of a function that contains the following. Label your axis. \( g(2)=1, \quad g(5)=-1, \quad \lim _{x \rightarrow 4} g(x)=-\infty \) \( \lim _{x \rightarrow 7^{-}} g(x)=\infty, \quad \lim _{x \rightarrow 7^{+}} g(x)=-\infty \)

To sketch the graph based on the given conditions, begin by drawing your x and y axes, labeling them accordingly. Plot the point at (2, 1) and the point at (5, -1). Next, as \(x\) approaches 4, the graph should plunge down towards \(-\infty\), indicating a vertical asymptote at \(x = 4\). Now for \(x = 7\), show two things: as \(x\) approaches 7 from the left (\(7^-\)), the function shoots up toward \(+\infty\) and as \(x\) approaches from the right (\(7^+\)), it dives down toward \(-\infty\). This creates a vertical asymptote at \(x = 7\). Connect these points smoothly, ensuring your graph reflects these behaviors for a clear, engaging visual of \(g(x)\). To spice up your understanding of graphing, imagine this function as a roller coaster: the steep drops represent the asymptotes, while the peaks at \(g(2) = 1\) and \(g(5) = -1\) are the exhilarating highs and lows of the ride! Each part of the graph tells a story of action, with the function dancing between valleys and mountains—just like a thrilling adventure! When sketching such functions, common mistakes include overlooking vertical asymptotes or not plotting key points accurately. Always check the limits to give your graph the right behavior at those critical points! Practice makes perfect, so don't hesitate to experiment with different shapes until you find one that fits the criteria seamlessly.