Problem 111 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The domain of the given function is \(x ∈ (-∞, -1) ∪ (5, +∞)\).

Step 1: 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: \(\frac{x+1}{x-5} > 0\).

Step 2: 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 \(x - 5 = 0\)).

Step 3: 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 \((-∞, -1)\), \((-1, 5)\), and \((5, +∞)\). 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.

Step 4: 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 \(x ∈ (-∞, -1) ∪ (5, +∞)\).