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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To graph the function \( f(x) = 5 + 2e^{-0.2x} \), we note that as \( x \) approaches infinity, \( e^{-0.2x} \) approaches 0, meaning \( f(x) \) approaches 5. Therefore, there is a horizontal asymptote at \( y = 5 \). As \( x \) approaches negative infinity, \( f(x) \) tends toward \( 5 + 2(1) = 7 \). Hence, the horizontal asymptote is \( y = 5 \). For the vertical asymptotes, we analyze the function's behavior. Since \( e^{-0.2x} \) is defined for all real numbers \( x \), our function does not have any values of \( x \) that make \( f(x) \) undefined. Thus, there are no vertical asymptotes. So, to summarize: The function has one horizontal asymptote, \( y = 5 \). C. There are no vertical asymptotes.
