Problem 59 - 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 the function is \((-2, 4) \cup (4, \infty)\).

Step 1: Understanding the Function

The function given is a logarithmic function: \( f(x) = \ln \left( \frac{x+2}{x-4} \right) \). The natural logarithm function, \( \ln(x) \), is only defined when the expression inside the logarithm is greater than zero.

Step 2: Set Inequality for Logarithm

To find where the function is defined, set the expression inside the logarithm greater than zero: \( \frac{x+2}{x-4} > 0 \). This inequality will help us determine the domain of the function.

Step 3: Determine when the Fraction is Positive

The fraction \( \frac{x+2}{x-4} \) is positive when both the numerator and the denominator are either both positive or both negative. Thus, we need to analyze the intervals determined by the zeros of the numerator (\( x = -2 \)) and the denominator (\( x = 4 \)).

Step 4: Test Intervals for Sign

Identify intervals based on the zeros: \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\). Test a value from each interval in \( \frac{x+2}{x-4} \).- For \((-\infty, -2)\), pick \(x = -3\): \( \frac{-3+2}{-3-4} = -\frac{1}{7} < 0\), so this interval is not in the domain.- For \((-2, 4)\), pick \(x = 0\): \( \frac{0+2}{0-4} = -\frac{1}{2} > 0\), so this interval is in the domain.- For \((4, \infty)\), pick \(x = 5\): \( \frac{5+2}{5-4} = 7/1 > 0\), so this interval is in the domain.

Step 5: Assemble the Domain

Combine the intervals that result in a positive expression: The domain is \((-2, 4) \cup (4, \infty)\), excluding \(x = -2\) and \(x = 4\) because the logarithm cannot be zero or undefined.

Key Concepts

Logarithmic Function

A logarithmic function is a powerful mathematical tool often used in various fields like mathematics, engineering, and the sciences. The function involved in this exercise is a logarithmic function with a natural logarithm expressed as \(f(x) = \ln \left( \frac{x+2}{x-4} \right)\). The key idea here is that the natural logarithm of a positive number results in a real number. However, if the argument inside the logarithm is less than or equal to zero, the function becomes undefined.
This behavior of the logarithmic function leads to finding the domain of the function by setting constraints on its input based on the rule that the natural logarithm only accepts positive numbers.

Inequalities

To explore the domain of logarithmic functions, understanding inequalities is crucial. In this exercise, we analyze the inequality \(\frac{x+2}{x-4} > 0\).

• Step 1: Identify the zero points of the numerator and the denominator, which are \(x = -2\) and \(x = 4\) respectively. These critical points helps us divide the real number line into intervals.
• Step 2: The intervals to test are \((−\infty, -2), (-2, 4), \text{and} (4, \infty)\). Selecting a test point from each interval helps in determining the sign of \(\frac{x+2}{x-4}\) in that interval.
• Step 3: If the test point results in a positive output, that interval is included in the domain. Only the intervals \((-2, 4)\) and \((4, \infty)\) captured this positive behavior.

Understanding inequalities allows us to solve and interpret problems like this, helping identify the regions where the function is defined.

Interval Notation

The domain of a function is often expressed using interval notation, a concise way to describe a set of numbers. Here, the purpose is to indicate where \(f(x)\) is defined.

• Open intervals \((-2, 4)\) and \((4, \infty)\) are used, implying that numbers within these ranges, but not including the endpoints, satisfy \( \ln \left(\frac{x+2}{x-4}\right) > 0\).
• A union symbol \(\cup\) connects these intervals, signifying that both sets of numbers form the entire domain.

This means that any \(x\) in the intervals \((-2, 4)\) or \((4, \infty)\) makes the function \(f(x)\) valid and result in real numbers.

Natural Logarithm

The natural logarithm, denoted by \(\ln(x)\), represents the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. It is a fundamental concept in mathematics that appears in various branches, including calculus and geometry.
In this context, the natural logarithm's defining property that it only accepts positive argument forms the basis of finding the function's domain. Since \(\ln(1) = 0\), and \(\ln(x)\) becomes undefined for non-positive numbers, the inequality \(\frac{x+2}{x-4} > 0\) reflects the constraints necessary for \(\ln\left(\frac{x+2}{x-4}\right)\) to provide real numbers.
The understanding of the natural logarithm ensures that one can solve equations and inequalities involving \(ln\) correctly, which is crucial for more advanced mathematical problem-solving.