Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=7 x e^{-0.5 x} \) Find the domain of \( f(x) \). 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 \( x= \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. The domain is all real \( x \)
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Beyond the Answer
The function \( f(x) = 7x e^{-0.5x} \) is defined for all real numbers since both the polynomial part \( 7x \) and the exponential part \( e^{-0.5x} \) can take any real values without restrictions. Thus, the graph will extend infinitely in both the positive and negative x-directions, showing that the function rises, peaks, and then descends towards zero as \( x \) increases. To illustrate this, when sketching the graph, you'll notice that the function crosses the x-axis at \( x = 0 \) and approaches \( y = 0 \) as \( x \) becomes large. Therefore, you can confidently select option B: The domain is all real \( x \).
