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

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**Step 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 real $$x$$, we have
$$
x^2 + 9 \ge 9 > 0.
$$
Thus, the inequality is satisfied for all real numbers, and no values need to be excluded.

**Answer (Domain):**  
The domain is all real numbers.

---

**Step 2. Find the $$ x $$-intercept(s)**

To find the $$ x $$-intercept(s), set $$ f(x) = 0 $$:
$$
\ln\left(x^2 + 9\right) = 0.
$$
Recall that $$ \ln(a) = 0 $$ if and only if $$ a = 1 $$. Therefore,
$$
x^2 + 9 = 1.
$$
Solve for $$ x $$:
$$
x^2 = 1 - 9 = -8.
$$
Since $$ x^2 = -8 $$ has no real solutions, the function has no $$ x $$-intercepts.

**Answer ($$ x $$-intercept):**  
There are no $$ x $$-intercepts.

---

**Step 3. Find the $$ y $$-intercept(s)**

The $$ y $$-intercept occurs when $$ x = 0 $$:
$$
f(0) = \ln\left(0^2 + 9\right) = \ln(9).
$$
Approximating $$ \ln(9) $$:
$$
\ln(9) \approx 2.2 \quad \text{(to one decimal place)}.
$$

**Answer ($$ y $$-intercept):**  
The $$ y $$-intercept is $$ y \approx 2.2 $$.

---

**Summary of Pertinent Information:**

1. **Domain:**  
   The domain is all real numbers.

2. **$$ x $$-intercepts:**  
   There are no $$ x $$-intercepts.

3. **$$ y $$-intercept:**  
   The $$ y $$-intercept is $$ y \approx 2.2 $$.

The graph of
$$
f(x) = \ln\left(x^2 + 9\right)
$$
will show a curve that is defined for all $$ x $$, has no $$ x $$-intercepts, and crosses the $$ y $$-axis at approximately $$(0, 2.2)$$.
**Summary:** - **Domain:** All real numbers. - **$$ x $$-intercepts:** None. - **$$ y $$-intercept:** Approximately $$ y = 2.2 $$. The graph of $$ f(x) = \ln(x^2 + 9) $$ is defined for all real numbers, has no $$ x $$-intercepts, and crosses the $$ y $$-axis at $$ y \approx 2.2 $$.

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