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What is the equation of the vertical asymptote of the func \( f(x)=\log _{9}(x+6) \) ?

Ask by Harper Flynn. in the United States
Jan 25,2025

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Answer

The vertical asymptote is at \( x = -6 \).

Solution

To determine the vertical asymptote of the function \( f(x) = \log_{9}(x + 6) \), we need to identify the value of \( x \) that makes the argument of the logarithm equal to zero. A logarithmic function \( \log_b(z) \) is undefined when \( z \leq 0 \). Therefore, the vertical asymptote occurs where the argument \( x + 6 = 0 \). **Steps to Find the Vertical Asymptote:** 1. **Set the argument equal to zero:** \[ x + 6 = 0 \] 2. **Solve for \( x \):** \[ x = -6 \] 3. **Conclusion:** The function approaches negative infinity as \( x \) approaches \(-6\) from the right, and it is undefined for \( x \leq -6 \). Thus, the vertical asymptote is the vertical line at \( x = -6 \). **Final Answer:** \[ x = -6 \]

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To find the vertical asymptote of the function \( f(x) = \log_{9}(x+6) \), we analyze when the argument of the logarithm becomes zero, since logarithms are undefined for non-positive values. Setting the argument \( x + 6 = 0 \), we find \( x = -6 \). Thus, the vertical asymptote occurs at \( x = -6 \), which means as \( x \) approaches -6 from the right, \( f(x) \) will decrease without bound. If you're trying to remember how to identify vertical asymptotes in logarithmic functions, just look for where the input of the log function equals zero! This is usually a straightforward method: find where the argument goes negative or zero, and voila, you’ve got your asymptote. So in this case, when \( x + 6 \) hits zero, you know you found it!

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