Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ \mathrm{f}(\mathrm{x})=7 \mathrm{xe} e^{-0.5 \mathrm{x}} $$. Find the domain of $$ \mathrm{f}(\mathrm{x} $$ ). 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 $$ \mathrm{x}= $$ (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 $$ \mathrm{f}(\mathrm{x}) $$. 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 $$ \mathrm{x}= $$ (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no x -intercepts.

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To analyze the function $$ f(x) = 7xe^{-0.5x} $$, we will follow these steps:

1. **Determine the domain of $$ f(x) $$**:
   - The function $$ f(x) $$ is composed of a polynomial term $$ 7x $$ and an exponential term $$ e^{-0.5x} $$.
   - The polynomial $$ 7x $$ is defined for all real numbers.
   - The exponential function $$ e^{-0.5x} $$ is also defined for all real numbers.
   - Therefore, the domain of $$ f(x) $$ is all real numbers.

2. **Find the x-intercepts of $$ f(x) $$**:
   - The x-intercepts occur where $$ f(x) = 0 $$.
   - Setting the function equal to zero:
     $$
     7xe^{-0.5x} = 0
     $$
   - This equation is satisfied if either $$ 7x = 0 $$ or $$ e^{-0.5x} = 0 $$.
   - The term $$ e^{-0.5x} $$ is never zero for any real $$ x $$, so we only need to solve $$ 7x = 0 $$:
     $$
     x = 0
     $$
   - Thus, the x-intercept is at $$ x = 0 $$.

3. **Sketch the graph of $$ f(x) $$**:
   - The function $$ f(x) $$ is a product of a linear function and an exponential decay function.
   - As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches 0 (since $$ 7x $$ becomes negative and $$ e^{-0.5x} $$ approaches infinity).
   - At $$ x = 0 $$, $$ f(0) = 0 $$.
   - As $$ x $$ increases, $$ f(x) $$ will initially increase due to the linear term, but will eventually decrease as the exponential decay dominates.
   - The function will have a maximum point before it starts to decrease.

Now, let's summarize the findings:

- **Domain**: The domain is all real $$ x $$. (Choice B)
- **X-intercept**: The x-intercept is at $$ x = 0 $$. (Choice A)

Next, I will sketch the graph of $$ f(x) $$.

### Graph Sketch
The graph of $$ f(x) = 7xe^{-0.5x} $$ will look like this:

- It starts from the origin (0,0).
- It increases to a maximum point and then decreases towards the x-axis as $$ x $$ increases.

Now, let's finalize the answers:

- **Domain**: B. The domain is all real $$ x $$.
- **X-intercept**: A. The x-intercept(s) is/are at $$ x = 0 $$.
- **Domain**: All real numbers. - **X-intercept**: $$ x = 0 $$.

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