Problem 61 - College Algebra

Question

What is the domain of the function \(f(x)=\ln \left(\frac{x+2}{x-4}\right) ?\) Discuss the result.

Step-by-Step Solution

Answer

The domain of \(f(x)\) is \((-\infty, -2) \cup (4, \infty)\).

Step 1: Understand the Function

The function given is a natural logarithm, which has specific domain restrictions. The logarithmic function \(\ln(x)\) is only defined for positive values of \(x\). Thus, the expression inside the logarithm must be greater than zero: \(\frac{x+2}{x-4} > 0\).

Step 2: Determine the Inequality

Set up the inequality derived from the function: \(\frac{x+2}{x-4} > 0\). This inequality will help us find the domain of the function, ensuring that the input of the logarithm function is positive.

Step 3: Analyze Critical Points

Find the critical points of the inequality. These occur when the numerator \((x+2)\) is zero and when the denominator \((x-4)\) is zero. Solving these gives \(x = -2\) and \(x = 4\) respectively. The function will be undefined at \(x = 4\), and it will change sign at both \(-2\) and \(4\).

Step 4: Test Intervals

The critical points \(-2\) and \(4\) divide the number line into three intervals: \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\). Test each interval to determine where the inequality \(\frac{x+2}{x-4} > 0\) holds. Choose test points from each interval: for example, \(x = -3\) for \((-\infty, -2)\), \(x = 0\) for \((-2, 4)\), and \(x = 5\) for \((4, \infty)\).

Step 5: Evaluate Inequality in Each Interval

- For \(x = -3\): \(\frac{-3+2}{-3-4} = \frac{-1}{-7} = \frac{1}{7} > 0\). So, the inequality holds for \((-\infty, -2)\).- For \(x = 0\): \(\frac{0+2}{0-4} = \frac{2}{-4} = -\frac{1}{2} < 0\). The inequality does not hold for \((-2, 4)\).- For \(x = 5\): \(\frac{5+2}{5-4} = \frac{7}{1} = 7 > 0\). The inequality holds for \((4, \infty)\).

Step 6: Conclusion About Domain

The logarithm function being positive only holds for values in \((-\infty, -2) \cup (4, \infty)\). These intervals are the valid domain of the function \(f(x) = \ln \left(\frac{x+2}{x-4}\right)\). The function is not defined at \(x=-2\) or \(x=4\) as these cause the numerator or denominator to be zero, making the fraction zero or undefined.

Key Concepts

Natural logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithmic function based on the constant \( e \), where \( e \approx 2.718 \). It's crucial to remember that the argument of a natural logarithm must be strictly positive, i.e., \( x > 0 \). This is because the logarithmic function is undefined for non-positive values. When you see a natural logarithm, like in our problem with \( \ln\left(\frac{x+2}{x-4}\right) \), it immediately hints at the need to ensure the expression inside the logarithm is greater than zero. This requirement frames our entire approach to finding the domain of the function. The positivity requirement will guide our steps in setting up an inequality to isolate where the function is valid.

Inequality solving

Inequality solving is the process of finding the values of \( x \) that satisfy a particular inequality. In our case, we set up the inequality \( \frac{x+2}{x-4} > 0 \), based on the need for the expression inside the natural logarithm to be positive. Solving this inequality involves a few steps.

• Identify the critical points by solving \( x+2 = 0 \) and \( x-4 = 0 \). This gives us \( x = -2 \) and \( x = 4 \).
• These points are critical as they indicate where the expression could potentially change from positive to negative or vice versa.
• Recognize that \( x = 4 \) will make the denominator zero, leading to an undefined expression, hence it is not part of the solution set.

Therefore, the inequality leads us to test the intervals created by these critical points to find where the inequality holds true.

Critical points

Critical points, in the context of inequalities, are the values of \( x \) where the function changes its sign. For rational expressions like \( \frac{x+2}{x-4} \), these points are derived from the numerator and denominator.

• The numerator gives a critical point at \( x = -2 \), where the function equals zero.
• The denominator gives another at \( x = 4 \), where the function becomes undefined.

These points are not within the domain but are vital in dividing the number line for further analysis. Around these critical points, we then test the behavior of the inequality in separate intervals. Note how these points help in ensuring both positivity of the logarithmic argument and defining our domain boundaries.

Interval testing

Interval testing is a method used to check where an inequality holds between its critical points. After identifying \(-2\) and \(4\) as critical points, these divide our number line into intervals: \((-\infty, -2), (-2, 4), \text{and } (4, \infty)\).By selecting a test point from each interval:

• For \((-\infty, -2)\), test \( x = -3 \): \( \frac{-3+2}{-3-4} = \frac{1}{7} \) is positive, so this interval works.
• For \((-2, 4)\), try \( x = 0 \): \( \frac{0+2}{0-4} = -\frac{1}{2} \) is negative, excluding this interval.
• For \((4, \infty)\), use \( x = 5 \): \( \frac{7}{1} = 7 \) is positive, meaning this interval works.

Thus, the solution set is \((-\infty, -2) \cup (4, \infty)\). This approach is systematic and confirms which regions meet the inequality conditions, thereby establishing the domain of our function.