Summarize the pertinent information obtained by applyirg the graphing stialegy and skotch the graph of $$ \mathrm{y}=\mathrm{f}(\mathrm{x}) $$. $$ f(x)=\ln (x+3)-2 $$ Select the correct choice below and, if necessary, fill in the answer box to completo your choice. A. The doman of $$ f $$ is $$ (-3, \infty) $$ (Type your answer in interval notation. Use a comma lo separate answers as needed.) B. The domain of $$ f $$ is empty Select the correct choice below and, if necessary, fill in the answer box to complote your choice. A. The $$ x $$-intercept(s) oif is $$ x= $$ $$ \square $$ . (Type an exact answer. Use a comma lo separale answers as needed.) B. The function I has no $$ x $$-intercepts.

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UpStudy AI · Accepted Answer

1. Determine the domain of $$ f(x) = \ln(x+3)-2 $$:
   - The argument of the logarithm, $$ x+3 $$, must be positive:
     $$
     x+3 > 0 \quad \Rightarrow \quad x > -3.
     $$
   - Thus, the domain is:
     $$
     (-3, \infty).
     $$
     
2. Find the $$ x $$-intercept(s):
   - Set $$ f(x) = 0 $$ and solve:
     $$
     \ln(x+3)-2 = 0 \quad \Rightarrow \quad \ln(x+3)=2.
     $$
   - Exponentiate both sides to remove the logarithm:
     $$
     x+3 = e^2.
     $$
   - Solve for $$ x $$:
     $$
     x = e^2 - 3.
     $$
     
Domain Selection:  
A. The domain of $$ f $$ is $$ (-3, \infty) $$.

$$ x $$-Intercept Selection:  
A. The $$ x $$-intercept of $$ f $$ is $$ x=e^2-3 $$.
The domain of $$ f $$ is $$ (-3, \infty) $$ and the $$ x $$-intercept is $$ x = e^2 - 3 $$.

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