Summanize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)= \) In \( (x+3)-2 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) is decteasing on the subintorval(s) (Iype your answer in interval notation. Use a comma to separate answers as needed.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) has a local maximum at \( x= \) (Type an exact answer. Use a comma to separate answers as nceded.) B. The function f has no local maximuril. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function f has a local minimum at \( \mathrm{x}= \) (Type an exact answer. Use a comma to separate answers as needed.) B. The function f has no local minimum.

To analyze the function \( f(x) = \ln(x + 3) - 2 \), we will follow these steps: 1. **Determine the domain of the function.** 2. **Find the derivative to analyze increasing/decreasing behavior.** 3. **Identify critical points for local maxima and minima.** 4. **Sketch the graph based on the findings.** ### Step 1: Determine the Domain The function \( f(x) = \ln(x + 3) - 2 \) is defined when the argument of the logarithm is positive: \[ x + 3 > 0 \implies x > -3 \] Thus, the domain of \( f \) is \( (-3, \infty) \). ### Step 2: Find the Derivative To find the intervals where the function is increasing or decreasing, we need to compute the derivative \( f'(x) \): \[ f'(x) = \frac{d}{dx}(\ln(x + 3)) = \frac{1}{x + 3} \] The derivative \( f'(x) \) is positive when \( x + 3 > 0 \) (which is always true in the domain) and negative when \( x + 3 < 0 \) (not applicable here). Therefore, \( f'(x) > 0 \) for all \( x > -3 \). ### Step 3: Identify Critical Points Since \( f'(x) > 0 \) for all \( x > -3 \), there are no critical points where the derivative is zero. Thus, there are no local maxima or minima. ### Step 4: Summary of Findings - **Decreasing Intervals:** The function \( f \) is not decreasing on any interval. - **Local Maximum:** The function \( f \) has no local maximum. - **Local Minimum:** The function \( f \) has no local minimum. ### Final Answers 1. The function \( f \) is decreasing on the interval(s): **None** (or you can write it as an empty set). 2. The function \( f \) has a local maximum at \( x = \): **None**. 3. The function \( f \) has a local minimum at \( x = \): **None**. ### Sketch of the Graph The graph of \( f(x) = \ln(x + 3) - 2 \) will show a logarithmic curve starting from the point \( (-3, -2) \) and increasing without bound as \( x \) increases. The graph approaches the vertical asymptote at \( x = -3 \) and continues to rise as \( x \) moves to the right. If you need a visual representation, I can plot the graph for you. Would you like me to do that?