91Ó°ÊÓ

First, we need to determine the domain of the given functions to ensure we are graphing them only where they exist. The domain of \(\sqrt{x}\) is \(x \ge 0\), and likewise, the domain of \(\sqrt{x+1}\) is \(x \ge -1\). Since we are graphing in the interval \([0,4]\), both functions are defined and continuous on this interval.

Now, we proceed to find the x-intercepts (where \(y=0\)) and the y-intercepts (where \(x=0\)) for each function. i) For the function \(\sqrt{x}+1\), at \(x=0\), we get the y-intercept as \(y = \sqrt{0} + 1 = 1\). To find the x-intercept, set \(y = 0\): \[0 = \sqrt{x} + 1\] \[-1 = \sqrt{x}\] There is no real solution, as the square root function cannot give a negative result. Therefore, this function has no x-intercept. ii) For the function \(\sqrt{x+1}\), at \(x=0\), we get the y-intercept as \(y = \sqrt{0+1} = 1\). To find the x-intercept, set \(y = 0\): \[0 = \sqrt{x+1}\] \[0 = x+1\] \[-1 = x\] In this case, the x-intercept is \(x=-1\), but since we are only interested in the interval \([0,4]\), there is no x-intercept within the interval.

Now, we can plot the important points we found in Step 2 and start sketching the graphs. For the function \(\sqrt{x}+1\): 1. Plot the y-intercept \((0,1)\). 2. Recall that there is no x-intercept within the interval \([0,4]\). 3. Since the square root function increases on its domain, the graph of \(\sqrt{x}+1\) will be a simple translation 1 unit up from the graph of \(\sqrt{x}\). 4. Sketch the graph using the points and information above. For the function \(\sqrt{x+1}\): 1. Plot the y-intercept \((0,1)\). 2. Recall that there is no x-intercept within the interval \([0,4]\). 3. The graph of \(\sqrt{x+1}\) will be a translation 1 unit to the left from the graph of \(\sqrt{x}\). 4. Sketch the graph using the points and information above. At the end of this process, you should have the graphs of both functions \(\sqrt{x}+1\) and \(\sqrt{x+1}\) on the interval [0,4].