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Chapter 4: Problem 111

Find the domain of each logarithmic function. $$f(x)=\log \left(\frac{x+1}{x-5}\right)$$

Short Answer

Expert verified
The domain of the given function is \(x 鈭� (-鈭�, -1) 鈭� (5, +鈭�)\).

Step by step solution

01

Set up the inequality

We know that for the function to be real, the value inside the logarithm must be greater than 0. So, we set up the following inequality: \(\frac{x+1}{x-5} > 0\).
02

Find the critical points

Critical points are the x-values that make the numerator or the denominator zero or undefined. The critical points, therefore, are \(x = -1\) (from \(x + 1 = 0\)) and \(x = 5\) (from \(x - 5 = 0\)).
03

Test the intervals

The critical points divide the number line into three intervals. We must test a number from each interval in the inequality to see if it produces a positive or negative result. The intervals are \((-鈭�, -1)\), \((-1, 5)\), and \((5, +鈭�)\). For the first interval, we can pick \(x = -2\). Subbing it into the inequality, we get a positive result, so this interval is included in the domain. For the second interval, we can pick \(x = 0\). Subbing it into the inequality, we get a negative result, so this interval is not included in the domain. For the third interval, we can pick \(x = 6\). Subbing it into the inequality, we get a positive result, so this interval is included in the domain.
04

Write the final answer

The domain of the function \(f(x) = \log \left(\frac{x+1}{x-5}\right)\) is the combination of the intervals that resulted in a positive output when tested in the inequality. So, the domain is \(x 鈭� (-鈭�, -1) 鈭� (5, +鈭�)\).

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