Summarize the pert nent information obtained by applying the graphing strategy and sketch the graph of $$ \mathrm{y}=\mathrm{f}(\mathrm{x}) $$. $$ f(x)=\ln (x+8)-1 $$ Select the correct choice below and, il necessary, fill in the answer box to complete your choice. A. The $$ x $$-intercept(s) of $$ f $$ is $$ x=e-B $$. (Type an exact answer. Use a comma to soparate answers as needed.) B. The function I has no $$ x $$-intercepls. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $$ y $$-intercept of $$ f $$ is $$ y=\ln (8)-1 $$. (Type an exact answer. Use a comma to separate answers as needed.) B. The function $$ f $$ has no $$ y $$-intercept. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function $$ f $$ is increasing on the subinterval(s) $$ \square $$ . (Type your answer in interval notation. Use a comma to separalo answers as needed.) B. The function $$ f $$ is never increasing.

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**Step 1. Find the domain.**  
For the logarithm $$ \ln(x+8) $$ to be defined, we must have  
$$
x+8>0, \quad \text{or} \quad x>-8.
$$  
Thus, the domain of $$ f $$ is  
$$
(-8, \infty).
$$

**Step 2. Find the $$ x $$-intercept.**  
The $$ x $$-intercept occurs where $$ f(x)=0 $$:
$$
\ln(x+8)-1=0.
$$
Add $$ 1 $$ to both sides:
$$
\ln(x+8)=1.
$$
Exponentiate both sides:
$$
x+8 = e.
$$
Solve for $$ x $$:
$$
x=e-8.
$$
The correct choice is:  
A. The $$ x $$-intercept(s) of $$ f $$ is $$ x=e-8 $$.

**Step 3. Find the $$ y $$-intercept.**  
The $$ y $$-intercept is found by evaluating $$ f(0) $$:
$$
f(0)=\ln(0+8)-1=\ln(8)-1.
$$
The correct choice is:  
A. The $$ y $$-intercept of $$ f $$ is $$ y=\ln(8)-1 $$.

**Step 4. Determine where $$ f $$ is increasing.**  
Differentiate $$ f(x) $$:
$$
f'(x)=\frac{d}{dx}\left[\ln(x+8)-1\right]=\frac{1}{x+8}.
$$
Since $$ x+8>0 $$ for $$ x>-8 $$ (the domain of $$ f $$), we have  
$$
f'(x)>0 \quad \text{for } x\in (-8,\infty).
$$
Thus, the function is increasing on  
$$
(-8,\infty).
$$
The correct choice is:  
A. The function $$ f $$ is increasing on the subinterval $$ (-8,\infty) $$.

**Graphing Summary:**  
- The domain of $$ f $$ is $$ (-8,\infty) $$ with a vertical asymptote at $$ x=-8 $$.  
- The $$ x $$-intercept is at $$ x=e-8 $$.  
- The $$ y $$-intercept is at $$ y=\ln(8)-1 $$.  
- The function is increasing on the interval $$ (-8,\infty) $$.

A sketch of the graph would show the curve approaching the vertical asymptote as $$ x\to -8^+ $$, crossing the $$ x $$-axis at $$ x=e-8 $$, and rising continuously for $$ x>-8 $$.
**Summary:** - **Domain:** $$ x > -8 $$ - **$$ x $$-intercept:** $$ x = e - 8 $$ - **$$ y $$-intercept:** $$ y = \ln(8) - 1 $$ - **Increasing Interval:** $$ (-8, \infty) $$ **Graph:** The function $$ f(x) = \ln(x+8) - 1 $$ has a vertical asymptote at $$ x = -8 $$, crosses the $$ x $$-axis at $$ x = e - 8 $$, and increases continuously for all $$ x > -8 $$.

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