Question

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

Explanation:

Step1: Find the domain

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 since \(x^{2}\) is defined for all real \(x\), the domain of \(y = 9e^{-0.5x^{2}}\) is all real \(x\).

Step2: Find the \(x -\)intercepts

Set \(y = 0\), so \(9e^{-0.5x^{2}}=0\).
We know that \(e^{-0.5x^{2}}=\frac{1}{e^{0.5x^{2}}}\), and \(e^{0.5x^{2}}>0\) for all real \(x\). Then \(9e^{-0.5x^{2}}>0\) for all real \(x\). So there are no \(x -\)intercepts.

Step3: Find the \(y -\)intercepts

Set \(x = 0\), then \(y=9e^{-0.5\times0^{2}}\).
Since \(e^{0}=1\), we have \(y = 9\times1=9\).

Answer:

For the domain: B. The domain is all real \(x\).
For the \(x -\)intercepts: B. There are no \(x -\)intercepts.
For the \(y -\)intercepts: A. The \(y -\)intercept(s) is/are at \(y = 9\).