Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ \mathrm{f}(\mathrm{x})=1+3 e^{-0.2 \mathrm{x}} $$ Find any horizontal asymptotes of $$ \mathrm{f}(\mathrm{x}) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is (Type equations.) C. There are no horizontal the bottom asymptotes. Find any vertical asymptotes of $$ \mathrm{f}(\mathrm{x}) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $$ \square $$ and the rightmost asymptote is (Type equations.) C. There are no vertical asymptotes.

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To find the horizontal asymptotes of the function $$ f(x) = 1 + 3e^{-0.2x} $$, we need to determine the behavior of the function as $$ x $$ approaches positive or negative infinity.

1. **Horizontal Asymptotes:**
   - To find the horizontal asymptotes, we need to consider the behavior of the function as $$ x $$ approaches positive or negative infinity.
   - As $$ x $$ approaches positive infinity, the term $$ e^{-0.2x} $$ approaches 0.
   - Therefore, the function $$ f(x) = 1 + 3e^{-0.2x} $$ approaches 1 as $$ x $$ approaches positive infinity.
   - Hence, the function has one horizontal asymptote at $$ y = 1 $$.

2. **Vertical Asymptotes:**
   - To find the vertical asymptotes, we need to determine if there are any values of $$ x $$ for which the function is undefined.
   - The function $$ f(x) = 1 + 3e^{-0.2x} $$ is defined for all real values of $$ x $$ since the exponential function $$ e^{-0.2x} $$ is always positive.
   - Therefore, there are no vertical asymptotes for the function $$ f(x) = 1 + 3e^{-0.2x} $$.

Based on the analysis, the correct choices are:
- For horizontal asymptotes: A. The function has one horizontal asymptote, $$ y = 1 $$.
- For vertical asymptotes: C. There are no vertical asymptotes.
The function $$ f(x) = 1 + 3e^{-0.2x} $$ has one horizontal asymptote at $$ y = 1 $$ and no vertical asymptotes.

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