Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=\frac{x^{2}+9 x+18}{x^{2}+6 x+9} \) Find the intervals where \( f(x) \) is increasing or decreasing. Select the correct choice below and fill in the answer box(es) to complete your choice. A. The function is increasing on \( \square \). It is never decreasing. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is increasing on \( \square \). It is decreasing on \( \square \). (Type exact answers. Type your answers in interval notation. Use commas to separate answers as needed.) C. The function is decreasing on \( \square \). It is never increasing. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.)
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The Deep Dive
To analyze the function \( f(x) = \frac{x^{2}+9x+18}{x^{2}+6x+9} \), we can first factor the numerator and denominator. The numerator factors into \( (x+3)(x+6) \), while the denominator factors into \( (x+3)^{2} \). Next, we simplify \( f(x) \): \[ f(x) = \frac{(x+3)(x+6)}{(x+3)^{2}} = \frac{x+6}{x+3}, \quad x \neq -3. \] We find the derivative \( f'(x) = \frac{(1)(x+3) - (x+6)(1)}{(x+3)^{2}} = \frac{-3}{(x+3)^{2}} \). Since the numerator is negative and the denominator is always positive (except at \( x=-3 \), where the function is undefined), \( f'(x) < 0 \) for \( x \neq -3 \). Hence, the function is decreasing for all \( x \) except \( x = -3 \) where it does not exist. **Answer Choices:** B. The function is increasing on \( (-\infty, -3) \) and decreasing on \( (-3, \infty) \).
