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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.

Ask by Sanders George. in the United States
Mar 17,2025

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**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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The function \( f(x) = 5 + 2 e^{-0.2x} \) is a transformation of the exponential decay function. As \( x \to \infty \), \( e^{-0.2x} \) approaches 0, making \( f(x) \) approach 5. For \( x = 0 \), \( f(0) = 5 + 2(1) = 7 \). Thus, the graph approaches a horizontal asymptote at \( y = 5 \) and starts at \( (0, 7) \). The domain of \( f(x) \) is all real numbers \( x \), so the correct choice is **B. The domain is all real x.** Since \( f(x) \) will never be zero, there are no \( x \)-intercepts, hence **B. There are no x-intercepts.** For the \( y \)-intercept, we find \( f(0) = 7 \), leading us to **A. The y-intercept(s) is/are at \( y = 7 \).**

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