91Ó°ÊÓ

First, we need to find the domain and range of the function. The domain is the set of all possible x-values that can be input into the function while the range is the set of all possible y-values that can be output from the function. For the given function \(y = \frac{2}{x}\), we can see that x cannot be equal to 0, as division by zero is undefined. Therefore, the domain of the function is \(x \neq 0\) or \((-∞, 0) \cup (0, ∞)\). For every nonzero value of x, there will be a corresponding y-value in the function. Therefore, the range of the function is \(y \neq 0\) or \((-∞, 0) \cup (0, ∞)\).

Next, we need to analyze how the function behaves as x approaches zero from both the positive and negative sides. As \(x \rightarrow 0^+\), \(y = \frac{2}{x} \rightarrow ∞\), which means the graph goes to infinity as x approaches zero from the right. As \(x \rightarrow 0^-\), \(y = \frac{2}{x} \rightarrow -∞\), which means the graph goes to negative infinity as x approaches zero from the left. This tells us that there are vertical asymptotes at \(x = 0\), and the graph will never touch the y-axis. Similarly, as y approaches zero from both positive and negative sides: As \(y \rightarrow 0^+\), \(x = \frac{2}{y} \rightarrow ∞\), which means the graph goes to infinity as y approaches zero from the top. As \(y \rightarrow 0^-\), \(x = \frac{2}{y} \rightarrow -∞\), which means the graph goes to negative infinity as y approaches zero from the bottom. This tells us that there is a horizontal asymptote at \(y = 0\), and the graph will never touch the x-axis.

Now, we will plot some key points on the graph to better understand the shape of the graph. We can choose some significant x-values and find their corresponding y-values using the given equation. For example, we can choose \(x = -2, -1, 1, 2\). For \(x = -2\), \(y = \frac{2}{-2} = -1\). For \(x = -1\), \(y = \frac{2}{-1} = -2\). For \(x = 1\), \(y = \frac{2}{1} = 2\). For \(x = 2\), \(y = \frac{2}{2} = 1\). The points are (-2, -1), (-1, -2), (1, 2), and (2, 1).

Now that we have some key points and know the behavior of the function around the x and y axes, we can sketch the graph. Draw the vertical asymptote at \(x = 0\) and the horizontal asymptote at \(y = 0\). Plot the points (-2, -1), (-1, -2), (1, 2), and (2, 1) on the graph. As x gets closer to 0, the graph will approach the asymptotes but never touch them. Also, the graph extends to positive and negative infinity along both axes. Connect the points and draw two separate curves in each quadrant. The curve in the first quadrant extends from the horizontal asymptote on the right to the vertical asymptote on the top. The curve in the third quadrant extends from the horizontal asymptote on the left to the vertical asymptote on the bottom. The final sketch should show the graph of the function \(y = \frac{2}{x}\) in the first and third quadrants, with asymptotes at \(x = 0\) and \(y = 0\).