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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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 \( y = 7 \).**