Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=lnleft(x^{2}+36
ight) ).
find the domain. 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= )
(use a comma to separate answers as needed.)
b. the domain is all real numbers.
find the ( x )-intercept(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the ( x )-intercept(s) is (are) ( x= )
(round to one decimal place as needed. use a comma to separate answers as needed.)
b. there are no ( x )-intercepts.

Explanation:

Step1: Find the domain

For the function \(y = \ln(u)\), the argument \(u=x^{2}+36\) must satisfy \(u>0\).
Since \(x^{2}\geqslant0\) for all real \(x\), then \(x^{2}+36\geqslant36>0\) for all real \(x\).

Step2: Find the \(x -\) intercept

The \(x -\) intercepts occur when \(y = f(x)=0\).
Set \(\ln(x^{2}+36)=0\).
Using the property \(\ln a = b\Leftrightarrow a = e^{b}\), we have \(x^{2}+36=e^{0}=1\).
Then \(x^{2}=1 - 36=- 35\).
Since \(x^{2}=-35\) has no real solutions (because the square of a real number \(x\), \(x^{2}\geqslant0\) for all real \(x\)).

Answer:

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