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 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. 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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**Step 1. Domain** The function is $$ f(x)=5+2e^{-0.2x}. $$ Since the exponential function $$e^{-0.2x}$$ is defined for all real numbers, the domain of $$f(x)$$ is all real numbers. **Answer (Domain):** B. The domain is all real $$x$$. --- **Step 2. X-intercepts** To find the x-intercepts, set $$f(x)=0$$: $$ 5+2e^{-0.2x}=0. $$ Subtract 5 from both sides: $$ 2e^{-0.2x}=-5. $$ Since $$e^{-0.2x}$$ is always positive, the equation $$2e^{-0.2x}=-5$$ has no solutions. **Answer (X-intercepts):** B. There are no x-intercepts. --- **Step 3. Y-intercepts** To find the y-intercept, evaluate $$f(x)$$ at $$x=0$$: $$ f(0)=5+2e^{-0.2(0)}=5+2e^0=5+2(1)=7. $$ So, the y-intercept is at $$(0,7)$$. **Answer (Y-intercepts):** A. The y-intercept is at $$y=7$$. --- **Additional Pertinent Information from the Graphing Strategy** - **Behavior as $$x \to +\infty$$:** As $$x \to +\infty$$, $$ e^{-0.2x} \to 0 \quad \text{and} \quad f(x)\to5. $$ Therefore, there is a horizontal asymptote at $$y=5$$. - **Behavior as $$x \to -\infty$$:** As $$x \to -\infty$$, $$ e^{-0.2x} \to +\infty \quad \text{and} \quad f(x)\to+\infty. $$ - **General Shape:** The graph of $$f(x)$$ is an exponential decay shifted upward by 5. It starts at high values for large negative $$x$$, passes through the y-intercept $$(0,7)$$, and gradually decreases toward the horizontal asymptote $$y=5$$ as $$x$$ increases. --- **Sketch Description** - The graph has no vertical asymptotes. - It is always above the horizontal asymptote $$y=5$$ (with $$f(x)<7$$ for $$x>0$$ and $$f(x)>7$$ for $$x<0$$). - The smooth and continuous curve approaches $$y=5$$ as $$x \to +\infty$$ and rises steeply as $$x \to -\infty$$. All pertinent information has been summarized and the answers have been provided above. **Summary:** - **Domain:** All real numbers. - **X-intercepts:** None. - **Y-intercept:** At $$y=7$$. **Graph Description:** - The graph is an exponential decay curve shifted upward by 5. - It approaches a horizontal asymptote at $$y=5$$ as $$x$$ increases. - It has no vertical asymptotes and no x-intercepts. - The y-intercept is at $$(0,7)$$. **Additional Notes:** - As $$x$$ approaches positive infinity, $$f(x)$$ approaches 5. - As $$x$$ approaches negative infinity, $$f(x)$$ increases without bound. - The graph is always above the horizontal asymptote $$y=5$$.

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