Question
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Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( \mathrm{f}(\mathrm{x})=\ln \left(\mathrm{x}^{2}+9\right) \). Find vertical asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The vertical asymptote(s) is (are) \( x= \) (Use a comma to separate answers as needed.) B. There are no vertical asymptotes. Find horizontal asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The horizontal asymptote(s) is (are) \( y=\square \) (Use a comma to separate answers as needed.) B. There are no horizontal asymptotes.

Ask by Cook Barrett. in the United States
Mar 17,2025

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Answer

There are no vertical or horizontal asymptotes for the function \( f(x) = \ln(x^2 + 9) \).

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The Deep Dive

To find the vertical asymptotes of the function \( \mathrm{f}(\mathrm{x})=\ln \left(\mathrm{x}^{2}+9\right) \), we first examine the expression inside the logarithm. Since \( \mathrm{x}^{2}+9 \) is always positive for all real \( x \) (as \( \mathrm{x}^{2} \) must be non-negative and adding 9 makes it strictly positive), there are no values of \( x \) that cause the logarithm to be undefined. Therefore, there are no vertical asymptotes. For horizontal asymptotes, we analyze the behavior of \( \mathrm{f}(x) \) as \( x \) approaches positive and negative infinity. As \( x \) becomes very large or very small, \( \mathrm{x}^{2} \) dominates, and we have: \[ \mathrm{f}(x) \approx \ln(\mathrm{x}^{2}) = 2\ln|x| \] As \( x \) goes to both \( +\infty \) and \( -\infty \), \( \mathrm{f}(x) \) approaches \( +\infty \). Thus, there are no horizontal asymptotes. To summarize: A. The vertical asymptote(s) is (are) \( x= \) B. There are no vertical asymptotes. A. The horizontal asymptote(s) is (are) \( y= \) B. There are no horizontal asymptotes.

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