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 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 $$ x=\square $$ B. There are no x -intercepts. Find the $$ y $$-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 $$ y $$-intercept(s) is/are at $$ \mathrm{y}= $$ (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no $$ y $$-intercepts.

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**Domain**

The function is given by 
$$
f(x)=1+3e^{-0.2x}.
$$
Since the exponential function $$ e^{-0.2x} $$ is defined for every real $$ x $$, the domain is all real numbers.

Choice: **B. The domain is all real $$ x $$.**

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**$$x$$-Intercepts**

To find the $$ x $$-intercepts, we set $$ f(x)=0 $$ and solve:
$$
1+3e^{-0.2x}=0.
$$
Subtract $$ 1 $$ from both sides:
$$
3e^{-0.2x}=-1.
$$
Since $$ e^{-0.2x}>0 $$ for all $$ x $$ and multiplying by 3 keeps it positive, the left-hand side is always positive. Therefore, there is no solution.

Choice: **B. There are no $$ x $$-intercepts.**

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**$$y$$-Intercepts**

To find the $$ y $$-intercept, evaluate $$ f(x) $$ at $$ x=0 $$:
$$
f(0)=1+3e^{-0.2(0)}=1+3e^{0}=1+3=4.
$$
Thus, the $$ y $$-intercept is at $$ y=4 $$.

Choice: **A. The $$ y $$-intercept is at $$ y=4 $$.**

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**Summary of Graphing Information**

- **Domain:** All real numbers.
- **$$ x $$-Intercepts:** None.
- **$$ y $$-Intercept:** $$ (0,4) $$.
- **Behavior as $$ x \to \infty $$:** The exponential term $$ e^{-0.2x} \to 0 $$, so $$ f(x) \to 1 $$. This line $$ y=1 $$ is a horizontal asymptote.
- **Behavior as $$ x \to -\infty $$:** The exponential term $$ e^{-0.2x} \to +\infty $$, so $$ f(x) \to +\infty $$.
- **Overall Shape:** The function is a decreasing exponential curve starting from very high values (as $$ x \to -\infty $$), passing through the $$ y $$-intercept at $$ (0,4) $$, and approaching the horizontal asymptote $$ y=1 $$ as $$ x \to \infty $$.
**Summary:** - **Domain:** All real numbers. - **$$ x $$-Intercepts:** None. - **$$ y $$-Intercept:** $$ (0,4) $$. - **Asymptote:** $$ y=1 $$ as $$ x \to \infty $$. - **Behavior:** Decreasing exponential curve approaching $$ y=1 $$ from above as $$ x $$ increases, starting from $$ (0,4) $$ and increasing without bound as $$ x $$ decreases.

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