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Chapter 3: Problem 41

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$f(x)=-\log _{6}(x+2)$$

Short Answer

Expert verified
The domain of the function \(f(x)=-\log _{6}(x+2)\) is \((-2, +\infty)\). The x-intercept is -1 and the vertical asymptote is \(x=-2\). The graph is increasing, asymptotic to the vertical line \(x=-2\), crosses the x-axis at \(x=-1\), and is always below the x-axis.

Step by step solution

01

Find the Domain

The domain of \(f(x)\) is all \(x\) such that \(x+2>0\). Solving this inequality gives us \(x>-2\), so the domain of \(f(x)\) is \((-2, +\infty)\).
02

Find the X-intercept

The x-intercept is the value of \(x\) such that \(f(x) = 0\). Solving \(-\log _{6}(x+2) = 0\) will give us the x-intercept. As \(\log_a(a) = 1\), we can rewrite the equation to \(x+2 = 1\), which gives us \(x=-1\). So, -1 is the x-intercept.
03

Find the Vertical Asymptote

The vertical asymptote of a logarithmic function is the vertical line \(x=a\), where \(a\) is the value that makes the argument equal to zero. Here, the argument is \(x+2\), so solving \(x+2=0\) gives us \(x=-2\). Therefore, the vertical asymptote is \(x=-2\).
04

Sketch the Graph

With the obtained information, we can now sketch the graph. Remember: the graph is increasing and asymptotic to the vertical line \(x=-2\). It crosses the x-axis at \(x=-1\) and is always below the x-axis because the function is negative.

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