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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 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 \( x=\square \) 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 Wang Boone. in the United States
Mar 18,2025

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**Summary:** - **Domain:** All real numbers. - **\( x \)-Intercepts:** None. - **\( y \)-Intercept:** \( (0,4) \). - **Asymptote:** \( y=1 \) as \( x \to \infty \). - **Behavior:** Decreasing exponential curve approaching \( y=1 \) from above as \( x \) increases, starting from \( (0,4) \) and increasing without bound as \( x \) decreases.

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Bonus Knowledge

To summarize the graph \( f(x) = 1 + 3e^{-0.2x} \), we see that it is an exponential decay function shifted upwards by 1 unit. As \( x \) approaches infinity, \( f(x) \) approaches 1, while at \( x = 0 \), \( f(0) = 1 + 3(1) = 4 \). The domain of \( f(x) \) is all real numbers, so the answer is B. The \( x \)-intercepts occur when \( f(x) = 0 \), but solving \( 1 + 3e^{-0.2x} = 0 \) shows there are no solutions, making the answer B. The \( y \)-intercept is found at \( f(0) = 4 \), so the answer is A. The \( y \)-intercept is at \( y = 4 \).

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