Problem 106 - Precalculus

Question

Explain the mistake that is made. State the domain of the logarithmic function \(f(x)=\ln |x|\) in interval notation. Solution: since the absolute value eliminates all negative numbers, the domain is the set of all real numbers. Interval notation: \((-\infty, \infty)\) This is incorrect. What went wrong?

Step-by-Step Solution

Answer

The mistake was assuming the function is defined for all real numbers, including zero, which it is not. The correct domain is \((- inf, 0) \cup (0, inf)\).

Step 1: Understand the Function

The function given is \(f(x) = \ln |x|\). This is a logarithmic function where the natural logarithm is applied to the absolute value of \(x\). The natural logarithm function, \(\ln(x)\), is only defined for positive values of \(x\).

Step 2: Analyze the Effect of Absolute Value

The expression \(|x|\) takes any real number \(x\) and returns its absolute value, which is always non-negative. However, the natural logarithm \(\ln(x)\) is defined only for positive values, which implies \(|x| > 0\).

Step 3: Determine the Correct Domain

Since \(|x|\) must be positive for \(\ln|x|\) to be defined, \(x\) must be non-zero. Therefore, \(x\) can be any real number except 0, as \(|0| = 0\) and \(\ln(0)\) is undefined. Thus, the domain of \(f(x)\) in interval notation is \((-inf, 0) \cup (0, inf)\).

Step 4: Identify the Mistake

The solution's error was assuming that the function is defined for \(x = 0\) because \(|x|\) cannot equal zero in the domain of \(\ln |x|\). The statement that the domain is \((-inf, inf)\) is incorrect because it includes \(x = 0\), where the function is not defined.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we are identifying all the possible input values (usually represented as \(x\)) that allow the function to work without any issues. For a function to be defined, its domain must exclude values that might cause undefined behavior like division by zero or taking the logarithm of a negative number.

In the case of the natural logarithm function \(f(x) = \ln(x)\), it requires that \(x\) must be greater than zero. Therefore, whenever we have a function like \(f(x) = \ln|x|\), we need to ensure that the argument of the logarithm is strictly positive. The absolute value, \(|x|\), ensures non-negative values, but for the logarithm, it can only handle strictly positive values. Hence, the domain of \(f(x)\) becomes all real numbers except zero.

So, for \(f(x) = \ln|x|\), the set of all real numbers except zero is the domain, represented in interval notation as \((-\infty, 0) \cup (0, \infty)\). This captures all values except \(0\), where the natural logarithm would not be defined.

Absolute Value

The absolute value of a number \(x\), denoted as \(|x|\), is the non-negative value of \(x\) without regard to its sign. In simple terms, it measures how far a number is from zero.

Here’s how absolute value works:

• If \(x\) is positive or zero, then \(|x| = x\).
• If \(x\) is negative, then \(|x| = -x\), turning it into a positive number.

This property of turning negative numbers into positive is particularly useful because it means no matter what \(x\) is, \(|x|\) will always be zero or positive. This characteristic is crucial in functions like \(\ln|x|\), ensuring that the logarithmic function never deals with negative numbers but requiring that \(|x|\) still be positive (i.e., non-zero) to remain defined.

Interval Notation

Interval notation is a mathematical notation used to represent a set of values or a range of numbers, capturing the domain or range of a function in a neat and concise way. It uses parentheses \(()\) to denote open intervals or brackets \([]\) for closed intervals.

Some basics of interval notation include:

• Open intervals, such as \((a, b)\), include all numbers between \(a\) and \(b\) but not \(a\) and \(b\) themselves.
• Closed intervals, like \([a, b]\), include all numbers between \(a\) and \(b\), including \(a\) and \(b\).
• Semi-open or semi-closed intervals, such as \([a, b)\) or \((a, b]\), involve one end being inclusive and the other not.

When indicating a domain like \((-\infty, 0) \cup (0, \infty)\), this notation tells us we include all values between minus infinity and zero (not including zero), and from zero to positive infinity (again not including zero). This is a way to clearly express the solution to the earlier problem: while all real numbers are generally considered, the function must specifically exclude zero, as its value results in an undefined logarithm.