Question
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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 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 the bottom 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 (Type equations.) C. There are no vertical asymptotes.

Ask by Bates Coles. in the United States
Mar 18,2025

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Answer

The function \( f(x) = 1 + 3e^{-0.2x} \) has one horizontal asymptote at \( y = 1 \) and no vertical asymptotes.

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Beyond the Answer

To sketch the graph of \( f(x) = 1 + 3e^{-0.2x} \), we note that this function approaches a horizontal asymptote as \( x \to \infty \). As \( x \) increases, \( e^{-0.2x} \) approaches \( 0 \), leading to \( f(x) \to 1 \). Therefore, the horizontal asymptote is \( y = 1 \). The function has no vertical asymptotes, as it is defined for all real \( x \). **Horizontal Asymptote:** A. The function has one horizontal asymptote, \( y = 1 \). **Vertical Asymptote:** C. There are no vertical asymptotes.

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