Problem 49 - Precalculus

Question

Find the domain of the function. \(f(x)=\ln (7-x)+\ln (x-4)\)

Step-by-Step Solution

Answer

The domain of the function is \((4, 7)\).

Step 1: Understanding Logarithmic Functions

The function is composed of two logarithmic functions: \( \ln(7-x) \) and \( \ln(x-4) \). For the logarithm to be defined, the arguments must be positive: \( 7-x > 0 \) and \( x-4 > 0 \).

Step 2: Finding Constraints for Each Logarithm

For \( \ln(7-x) \) to be defined, \( 7-x > 0 \). Solving this inequality: \( x < 7 \). For \( \ln(x-4) \) to be defined, \( x-4 > 0 \). Solving this inequality: \( x > 4 \).

Step 3: Finding the Intersection of Solutions

The domain is determined by the intersection of the solutions to both inequalities. From \( x < 7 \) and \( x > 4 \), the domain of \( f(x) \) is \( 4 < x < 7 \).

Step 4: Expressing the Domain in Interval Notation

Write the domain as an interval: The function \( f(x) = \ln(7-x) + \ln(x-4) \) is defined for \( x \) in the interval \((4, 7)\).

Key Concepts

Understanding Logarithmic Inequalities

Logarithmic inequalities involve expressions where a logarithm function is compared to another value. Solving these inequalities works similarly to solving quadratic or linear inequalities. The primary rule is that the argument of a logarithmic function must be greater than zero— logarithms of non-positive numbers are undefined. For the function \(f(x)=\ln (7-x)+\ln (x-4)\), both \(7-x\) and \(x-4\) must be individually greater than zero to keep the function valid and solve the inequalities:

• \(7-x > 0\) implies \(x < 7\)
• \(x-4 > 0\) implies \(x > 4\)

This process of breaking down each part is essential to determine the valid range for \(x\) where both inequalities hold true at the same time. This ensures the logarithmic function's domain is correctly identified.

Converting to Interval Notation

Once you've solved the separate inequalities from the logarithmic function, the next step is to express the domain in a compact form, known as interval notation. This notation uses brackets to show that numbers within the specified range are included or excluded.For the function \(f(x)=\ln (7-x)+\ln (x-4)\), the inequalities \(x < 7\) and \(x > 4\) need to be combined. The solution to both gives us the interval \((4, 7)\). Note that:

• Parentheses, \(()\), are used to indicate that the endpoints are not included.
• In this example, both 4 and 7 are not part of the domain, only values between them.

Interval notation is a succinct way to communicate the domain, making it easy to read and understand the scope of valid inputs for the function.

Exploring the Intersection of Inequalities

Finding the intersection of inequalities like \(x<7\) and \(x>4\) essentially involves identifying the common values that satisfy both conditions. This means looking for a range where both inequalities overlap. To express the overlapping solution, imagine a number line:

• \(x<7\) indicates all numbers to the left of 7.
• \(x>4\) indicates all numbers to the right of 4.
• The intersection is the part of the line where both constraints hold true: \(4 < x < 7\).

This intersection defines the actual domain of the function \(f(x)\), ensuring that it stays within the realm of real and computational validity. Understanding how to find and verify the intersection of inequalities is a crucial skill in solving these kinds of problems effectively.