Question

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\frac{\sin \sqrt{x}}{\sqrt{x}} $$

Short answer

The graph of the function \(f(x) = \frac{\sin(\sqrt{x})}{\sqrt{x}}\) can be plotted using the viewing window \(x \in [0, 20], y \in [-0.5, 1.5]\). This window captures the oscillatory behavior and the point of interest at \(x=0\) where \(f(0)=1\). Note that there are multiple viewing windows that can capture the essence of the function, and our chosen window is one example.

Step-by-step solution

Find the domain

The domain of the function is the set of all \(x\) values for which \(f(x)\) is defined. Since the function involves a square root, we cannot have negative values within it, i.e., \(x \ge 0\). Also, we need to ensure that the denominator, \(\sqrt{x}\), is not equal to zero. In this case, the denominator is only zero when \(x = 0\), but as the function approaches \(x = 0\), it still has a well-defined limit. Therefore, the domain of the function is \(x \ge 0\).

Identify points of interest

At first, we can find the limit of the function as \(x\) approaches 0. We can apply L'Hôpital's Rule to compute the limit: \[ \lim_{x \to 0}\frac{\sin(\sqrt{x})}{\sqrt{x}} = \lim_{x \to 0}\frac{\frac{1}{2\sqrt{x}}\cos(\sqrt{x})}{\frac{1}{2\sqrt{x}}} = \lim_{x \to 0}\cos(\sqrt{x}) = \cos(0) = 1 \] We determine the function value at \(x = 0\) to be 1 by the L'Hôpital's rule, which is a point of interest. For the function \(f(x) = \frac{\sin(\sqrt{x})}{\sqrt{x}}\), its periodic oscillatory behavior comes from the \(\sin(\sqrt{x})\) term. As \(\sqrt{x}\) grows, the oscillations become more frequent. These insights hint that we should focus on the graph for small and mid-range \(x\) values while capturing the rising oscillatory behavior.

Choose an appropriate viewing window

Considering the previous steps, we choose an appropriate viewing window to capture the key aspects of the function without losing detail. A window with \(x\) between 0 and 20 and \(y\) between -0.5 and 1.5 should display the oscillatory behavior and the point of interest at \(x = 0\). So, we choose the viewing window: \(x \in [0, 20], y \in [-0.5, 1.5]\)

Plot the graph

Using the viewing window \([0,20] \times [-0.5,1.5]\), plot the graph of the function \(f(x) = \frac{\sin(\sqrt{x})}{\sqrt{x}}\). Observe its general oscilatory behavior and the fact that \(f(0) = 1\). Remember that there are multiple possible viewing windows that capture the essence of the function. The chosen window is one example.