Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ \mathrm{f}(\mathrm{x})=5+2 e^{-0.2 \mathrm{x}} $$. 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 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 $$ \square $$. (Type equations.) C. There are no vertical asymptotes.

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UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = 5 + 2 e^{-0.2 x} $$, we will follow these steps:

1. **Identify the behavior of the function as $$ x $$ approaches infinity and negative infinity** to find horizontal asymptotes.
2. **Check for vertical asymptotes** by determining where the function is undefined.

### Step 1: Horizontal Asymptotes

- As $$ x \to \infty $$:
  $$
  e^{-0.2 x} \to 0 \quad \text{(since the exponential function decays to zero)}
  $$
  Therefore,
  $$
  f(x) \to 5 + 2 \cdot 0 = 5
  $$

- As $$ x \to -\infty $$:
  $$
  e^{-0.2 x} \to \infty \quad \text{(since the exponential function grows)}
  $$
  Therefore,
  $$
  f(x) \to 5 + 2 \cdot \infty = \infty
  $$

Thus, the function has one horizontal asymptote at $$ y = 5 $$.

### Step 2: Vertical Asymptotes

To find vertical asymptotes, we look for values of $$ x $$ that make the function undefined. The function $$ f(x) = 5 + 2 e^{-0.2 x} $$ is defined for all real numbers $$ x $$ because the exponential function is defined everywhere.

Thus, there are no vertical asymptotes.

### Summary of Findings

- **Horizontal Asymptote**: The function has one horizontal asymptote at $$ y = 5 $$.
- **Vertical Asymptotes**: There are no vertical asymptotes.

### Final Answers

1. **Horizontal Asymptote**: 
   - A. The function has one horizontal asymptote, $$ y = 5 $$.

2. **Vertical Asymptotes**: 
   - C. There are no vertical asymptotes.

### Sketch of the Graph

The graph of $$ f(x) = 5 + 2 e^{-0.2 x} $$ will show the following characteristics:
- As $$ x $$ approaches infinity, the graph approaches the line $$ y = 5 $$.
- As $$ x $$ approaches negative infinity, the graph rises towards infinity.
- The graph is continuous and does not have any breaks or asymptotes in the vertical direction.

The graph will start from a high value when $$ x $$ is negative and will gradually decrease to approach the horizontal asymptote at $$ y = 5 $$.
The function $$ f(x) = 5 + 2 e^{-0.2 x} $$ has one horizontal asymptote at $$ y = 5 $$ and no vertical asymptotes.

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