Problem
Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $f(x)=2 x e^{-0.5 x}$. Find the domain of $f(x)$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is all real $x$, except $x=$. (Type an integer or a decimal. Use a comma to separate answers as needed.) B. The domain is all real $x$. Find the $x$-intercepts of $f(x)$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $x$-intercept(s) is/are at $x=\square$. (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no $x$-intercepts.
Solution
Step 1 ：The function $f(x)=2 x e^{-0.5 x}$ is defined for all real numbers. Therefore, the domain is all real x.
Step 2 ：The x-intercepts of a function are the points where the graph of the function crosses the x-axis. This happens when the function's value is zero. So, we need to solve the equation $f(x)=0$ to find the x-intercepts.
Step 3 ：Solving the equation $f(x)=0$ gives $x=0$. Therefore, the x-intercept of the function is at $x=0$.
Step 4 ：\(\boxed{\text{The domain of } f(x) \text{ is all real } x.}\)
Step 5 ：\(\boxed{\text{The } x\text{-intercept of } f(x) \text{ is at } x=0.}\)
