Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}+9}{\mathrm{x}-9} \). Find the intervals where \( \mathrm{f}(\mathrm{x}) \) is concave upward or downward. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is concave upward on B. The function is concave downward on is never concave downward. It is never concave upward. C. The function is concave upward on \( \square \). It is concave downward on \( \square \). Find the iocation of any inflection points of \( \mathrm{f}(\mathrm{x}) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is an inflection point at \( \mathrm{x}=\square \). (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no inflection points.
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To sketch the graph of \( f(x) = \frac{x+9}{x-9} \), we start by identifying key features. The vertical asymptote occurs at \( x = 9 \) (where the denominator is zero), and the horizontal asymptote is at \( y = 1 \) (as \( x \) approaches infinity). The function intersects the y-axis at \( (0, -1) \) and the x-axis at \( (-9, 0) \). The graph will approach these asymptotes but never cross them. Now, for the concavity, we find the second derivative of \( f(x) \) and analyze its sign. The function is concave upward when the second derivative is positive and concave downward when it is negative. After computation, we conclude that the function is concave upward on \( (-\infty, 9) \) and concave downward on \( (9, \infty) \). Finally, regarding inflection points, there is only one at \( x = 9 \), which is where the concavity changes. So in conclusion: A. The function is concave upward on \( (-\infty, 9) \). B. The function is concave downward on \( (9, \infty) \). A. There is an inflection point at \( x = 9 \).
