Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ \mathrm{f}(\mathrm{x})=\ln \left(\mathrm{x}^{2}+9\right) $$. Find vertical asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The vertical asymptote(s) is (are) $$ x= $$ (Use a comma to separate answers as needed.) B. There are no vertical asymptotes. Find horizontal asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The horizontal asymptote(s) is (are) $$ y=\square $$ (Use a comma to separate answers as needed.) B. There are no horizontal asymptotes.

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1. **Determine the Domain**

The function is 
   $$
   f(x)=\ln\left(x^2+9\right).
   $$
   For the logarithm to be defined, its argument must be positive:
   $$
   x^2+9>0.
   $$
   Since $$ x^2 \ge 0 $$ for all $$ x $$, it follows that
   $$
   x^2+9 \ge 9>0.
   $$
   Therefore, the domain of $$ f $$ is all real numbers, $$ (-\infty, \infty) $$.

2. **Vertical Asymptote(s)**

A vertical asymptote typically occurs when the function is undefined or tends to $$ \pm\infty $$ as $$ x $$ approaches a value at the boundary of the domain. Since $$ f(x) $$ is defined for all $$ x $$, and there is no value $$ x $$ for which $$ x^2+9 $$ equals zero, there are no vertical asymptotes.

*Vertical asymptote answer:*  
   B. There are no vertical asymptotes.

3. **Horizontal Asymptote(s)**

To check for horizontal asymptotes, consider the behavior of $$ f(x) $$ as $$ x $$ approaches $$ \pm \infty $$:
   
   For large $$|x|$$, 
   $$
   x^2+9 \approx x^2,
   $$
   so
   $$
   f(x) \approx \ln(x^2) = 2\ln|x|.
   $$
   As $$ |x| \to \infty $$, $$ \ln|x| \to \infty $$, and hence
   $$
   f(x) \to \infty.
   $$
   Therefore, there is no finite horizontal asymptote.

*Horizontal asymptote answer:*  
   B. There are no horizontal asymptotes.

4. **Sketching the Graph: Key Features**

- **Domain:** $$ (-\infty, \infty) $$.
   
   - **Even Function:** Since $$ f(x) = \ln\left(x^2+9\right) $$ is even ($$ f(x)=f(-x) $$), the graph is symmetric about the $$ y $$-axis.
   
   - **Minimum Value:** The function reaches its minimum at $$ x=0 $$:
     $$
     f(0)=\ln\left(0^2+9\right)=\ln(9).
     $$
   
   - **End Behavior:** As $$ |x| \to \infty $$, $$ f(x) \to \infty $$; the graph increases without bound.
   
   - **No Asymptotes:** There are no vertical or horizontal asymptotes as determined above.

The graph is a smooth curve that is symmetric about the $$ y $$-axis, with its lowest point at $$ (0,\ln9) $$ and rising slowly to $$ \infty $$ as $$ |x| $$ increases.

**Final Answers:**

- Vertical asymptote(s): B. There are no vertical asymptotes.
- Horizontal asymptote(s): B. There are no horizontal asymptotes.
There are no vertical or horizontal asymptotes for the function $$ f(x) = \ln(x^2 + 9) $$.

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