Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=7xe^{-0.5x} ).
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=square ).
(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 ( 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=square ).
(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.)

Explanation:

Domain

Step1: Analyze function form

The function \( f(x) = 7xe^{-0.5x}\) is a product of a polynomial \(y = 7x\) (domain \((-\infty,\infty)\)) and an exponential function \(y=e^{-0.5x}=\frac{1}{e^{0.5x}}\) (domain \((-\infty,\infty)\) since the denominator \(e^{0.5x}
eq0\) for all real \(x\)).

x - intercept

Step1: Set \(y = 0\)

Set \(f(x)=7xe^{-0.5x}=0\). Since \(e^{-0.5x}=\frac{1}{e^{0.5x}}>0\) for all \(x\in R\), then \(7xe^{-0.5x}=0\) implies \(x = 0\) (because if \(ab = 0\) and \(b
eq0\), then \(a = 0\), here \(a=x\) and \(b = 7e^{-0.5x}\)).

y - intercept

Step1: Set \(x = 0\)

Substitute \(x = 0\) into \(f(x)\): \(f(0)=7\times0\times e^{-0.5\times0}=0\) (using the rule \(a^0 = 1,a
eq0\), here \(a = e\)).

Answer:

• Domain: B. The domain is all real \(x\).
• x - intercept: A. The \(x\) - intercept(s) is/are at \(x = 0\).
• y - intercept: A. The \(y\) - intercept(s) is/are at \(y = 0\).