Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=\ln \left(x^{2}+64\right) $$ A. The domain is all real $$ x $$, except $$ x=\square $$ (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) $$ x=\square $$ (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. Find the domain**

For the function  
$$
f(x)=\ln\left(x^2+64\right),
$$  
the argument of the logarithm must be positive. That is, we require  
$$
x^2+64 > 0.
$$  
Since $$x^2$$ is nonnegative for all real $$x$$, and $$64$$ is positive, it follows that  
$$
x^2+64\geq64>0
$$  
for all real $$x$$. Therefore, the domain is:

B. The domain is all real numbers.

---

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

The $$x$$-intercepts are obtained by setting $$f(x)=0$$. That gives:  
$$
\ln\left(x^2+64\right)=0.
$$
Recall that $$\ln(y)=0$$ if and only if $$y=1$$. Thus, we set:  
$$
x^2+64=1.
$$
Solving for $$x$$, we have:  
$$
x^2 = 1-64 = -63.
$$
Since $$x^2$$ cannot be negative for real numbers, there is no solution. Therefore:

B. There are no $$x$$-intercepts.

---

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

The $$y$$-intercept is found by setting $$x=0$$. Substitute into the function:  
$$
f(0)=\ln\left(0^2+64\right)=\ln(64).
$$
Numerically,  
$$
\ln(64) \approx 4.2 \quad (\text{to one decimal place}).
$$
Thus, the $$y$$-intercept is:

A. The $$y$$-intercept(s) is (are) $$ y \approx 4.2 $$.

---

**Summary and Graph Sketch Information**

- **Domain:** All real numbers.
- **$$x$$-intercepts:** None.
- **$$y$$-intercept:** $$ (0, \ln(64)) $$ or approximately $$(0, 4.2)$$.

For sketching the graph:
- The graph of $$\ln\left(x^2+64\right)$$ is symmetric about the $$y$$-axis because the expression $$x^2+64$$ is even in $$x$$.
- As $$|x|$$ increases, $$x^2+64$$ increases, and the logarithm function slowly increases.
- The graph has a minimum at $$x=0$$ with value $$\ln(64)$$.

This completes the analysis and sketching strategy for $$ f(x)=\ln\left(x^2+64\right) $$.
**Summary:** - **Domain:** All real numbers. - **$$x$$-intercepts:** None. - **$$y$$-intercept:** Approximately $$4.2$$ at $$x=0$$. **Graph Sketch:** - The graph is symmetric about the $$y$$-axis. - It has a minimum at $$x=0$$ with a value of approximately $$4.2$$. - As $$|x|$$ increases, the graph slowly rises. This completes the analysis and sketching strategy for $$ f(x)=\ln\left(x^2+64\right) $$.

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