summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of (…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=7xe^{-0.5x} ).\nfind the domain of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the domain is all real ( x ), except ( x=square ).\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. the domain is all real ( x ).\nfind the ( x )-intercepts of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( x )-intercept(s) is/are at ( x=square ).\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there are no ( x )-intercepts.\nfind the ( y )-intercepts of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( y )-intercept(s) is/are at ( y=square ).\n(type an integer or a decimal. use a comma to separate answers as needed.)

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=7xe^{-0.5x} ).\nfind the domain of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the domain is all real ( x ), except ( x=square ).\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. the domain is all real ( x ).\nfind the ( x )-intercepts of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( x )-intercept(s) is/are at ( x=square ).\n(type an integer or a decimal. use a comma to separate answers as needed.)\nb. there are no ( x )-intercepts.\nfind the ( y )-intercepts of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( y )-intercept(s) is/are at ( y=square ).\n(type an integer or a decimal. use a comma to separate answers as needed.)

Answer

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}\neq0) 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\neq0), 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\neq0), 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).