Summarize the pertinent information oblained 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 \( y \)-intercept of \( f \) is \( y= \) In (3) -2. B. The function \( f \) has no \( y \)-intercept Select the correct choico below and, if necessary, fill in the answer box to complete your choice A. The function \( f \) is increasing on the subinterval(s) ( -3 , oo) (Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function \( f \) is never increasing. Select the correct choice below and, if necessary, fill in the answer box to complote your choice. A. The function \( f \) is decieasing on the subinterval(s) (Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function \( f \) is never decreesing.

1. **Determine the Domain and Transformation:** The function is \[ f(x)=\ln(x+3)-2. \] Since the natural log is defined only for positive arguments, we have \[ x+3>0 \quad \Longrightarrow \quad x>-3. \] This means the domain of \( f \) is \((-3,\infty)\). The graph is a horizontal translation of the basic logarithm \( \ln(x) \) left by 3 units and a vertical translation downward by 2 units. 2. **Find the \( y \)-Intercept:** The \( y \)-intercept occurs when \( x=0 \). Evaluating: \[ f(0)=\ln(0+3)-2=\ln(3)-2. \] This shows that the \( y \)-intercept is \[ (0,\ln(3)-2). \] **Correct Choice:** A. The \( y \)-intercept of \( f \) is \( y=\ln(3)-2 \). 3. **Determine Where the Function is Increasing:** Compute the derivative: \[ f'(x)=\frac{d}{dx}[\ln(x+3)-2]=\frac{1}{x+3}. \] Since \( x>-3 \), the denominator \( x+3 \) is always positive, and hence \[ f'(x)>0 \quad \text{for } x\in(-3,\infty). \] This means the function \( f \) is increasing on \((-3,\infty)\). **Correct Choice:** A. The function \( f \) is increasing on the subinterval \((-3, \infty)\). 4. **Determine Where the Function is Decreasing:** As noted, the derivative \( f'(x) \) is positive on the entire domain \((-3,\infty)\), so the function is never decreasing. **Correct Choice:** B. The function \( f \) is never decreasing. 5. **Sketching the Graph:** - **Domain:** \( x>-3 \). - **Vertical Asymptote:** \( x=-3 \) (since \( \ln(x+3) \) is undefined at \( x=-3 \)). - **\( y \)-Intercept:** \( (0,\ln(3)-2) \). - **Behavior:** - The graph increases on the interval \((-3,\infty)\). - It is a shifted version of \( \ln(x) \): shifted left 3 units and down 2 units. - **Key Point:** - \( x \)-intercept can be found by solving \( \ln(x+3)-2=0 \): \[ \ln(x+3)=2 \quad \Longrightarrow \quad x+3=e^2 \quad \Longrightarrow \quad x=e^2-3. \] The summarized answers are: - \( y \)-intercept: \( y=\ln(3)-2 \). - Increasing on: \((-3,\infty)\). - Never decreasing.