91Ó°ÊÓ

We need to first find the domain and range of \(y = \ln{2x}\). Remember that you can only take the logarithm of a positive number. Thus, we have: Domain: \(2x > 0\) \(x > 0\) The domain is \(x > 0\), which means the graph will only exist for all positive x values. To find the range, we should note that the graph of the natural logarithm will always output all real numbers (i.e., from \(-\infty\) to \(\infty\)) when its domain is positive: Range: \((-\infty, \infty)\)

We will identify some key points on the graph to help us sketch the function. We can choose a few values of x within the domain and find their corresponding y values using the given equation. Let's try x values of \(x = \frac{1}{2}\), \(x = 1\), and \(x = \frac{3}{2}\): 1. For \(x = \frac{1}{2}\), \(y = \ln{(2 \cdot \frac{1}{2})} = \ln{1} = 0\) This gives us the point \((\frac{1}{2}, 0)\). 2. For \(x = 1\), \(y = \ln{(2 \cdot 1)} = \ln{2}\) This gives us the point \((1, \ln{2})\). 3. For \(x = \frac{3}{2}\), \(y = \ln{(2 \cdot \frac{3}{2})} = \ln{3}\) This gives us the point \((\frac{3}{2}, \ln{3})\).

Now that we have the domain, range, and key points, we can sketch the graph of \(y = \ln{2x}\): 1. Start by drawing the x-axis (horizontal) and y-axis (vertical). 2. Add tick marks at \(x = 1\), \(x = \frac{1}{2}\), and \(x = \frac{3}{2}\) on the x-axis. 3. Plot the key points: \((\frac{1}{2}, 0)\), \((1, \ln{2})\), and \((\frac{3}{2}, \ln{3})\). 4. Draw a smooth curve connecting the points that starts from the x-axis at the point \((\frac{1}{2}, 0)\) and goes through the other points. As x approaches infinity, the curve will continue to rise, and as x approaches 0, the curve will tend toward negative infinity. 5. Remember that the curve will not touch or cross the y-axis since the natural logarithm is undefined for a non-positive x. By following these steps, you should now have a rough sketch of the graph of \(y = \ln{2x}\).