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.

**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 \( x>-8 \).