Summarize the pertinent information oblained by applying the graphing strategy and sketch the graph of \( f(x)=7 x e^{-05 x} \) B. The domain is all real \( x \). Find the \( x \)-intercepts of \( f(x) \). 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 \( x=0 \) (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 \( f(x) \). 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 \( y= \) \( \square \) . (Type an integer or a decimal. Use a comma to separate answers as needed ) B. There are no \( y \)-intercepts.
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To sketch the graph of \( f(x) = 7x e^{-0.5x} \), you can start by noting that the function is a product of a linear term (\( 7x \)) and an exponential decay term (\( e^{-0.5x} \)), which generally shows that the function rises and then falls off. The domain is all real \( x \), meaning it stretches infinitely in both directions along the x-axis. The \( x \)-intercept occurs when \( f(x) = 0 \). Setting the equation \( 7x e^{-0.5x} = 0 \), we find that the only solution is \( x = 0 \). Thus, the \( x \)-intercept is at \( x = 0 \). For the \( y \)-intercept, substitute \( x = 0 \) into the function: \( f(0) = 7(0)e^{-0.5(0)} = 0 \). Consequently, the \( y \)-intercept is also at \( y = 0 \). Thus, both intercepts coincide at the origin!