Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=9e^{-0.5x^{2}} ).
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 ).
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= ) (type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no ( x )-intercepts.

Explanation:

Step1: Domain of the function

For the function \(y = 9e^{-0.5x^{2}}\), the exponential function \(e^{u}\) is defined for all real values of \(u\). Here \(u=-0.5x^{2}\), and \(x^{2}\) is defined for all real \(x\). So, the domain of \(f(x)\) is all real \(x\).

Step2: X - intercepts of the function

The \(x\) - intercepts of a function \(y = f(x)\) are found by setting \(y = 0\). So, we set \(9e^{-0.5x^{2}}=0\).
Since \(e^{-0.5x^{2}}=\frac{1}{e^{0.5x^{2}}}\), and \(e^{0.5x^{2}}>0\) for all real \(x\) (because the exponential function \(y = e^{v}\), where \(v = 0.5x^{2}\geq0\) and \(e^{v}>0\) for all \(v\in R\)), then \(9e^{-0.5x^{2}}>0\) for all real \(x\).

Answer:

For the domain: B. The domain is all real \(x\).
For the \(x\) - intercepts: B. There are no \(x\) - intercepts.