Problem 12 - Calculus of a Single Variable

Question

In Exercises 9–16, sketch the graph of the function and state its domain. $$ f(x)=\ln |x| $$

Step-by-Step Solution

Answer

The graph of \( f(x) = \ln |x| \) is a curve increasing towards the right for \( x > 0 \) and towards the left for \( x < 0 \) with a vertical asymptote at \( x = 0 \). The domain of the function is all x-values except for zero (i.e., \( x \neq 0 \)).

Step 1: Identify the Function Type

Recognize that this is a transformation of the basic logarithm function \( y = \ln x \), where the input has an absolute value function applied. The absolute value function will reflect any negative x-values over the y-axis, effectively 'folding' the graph over the y-axis.

Step 2: Determine the Domain of the Function

The domain of a function is the set of x-values for which the function is defined. Since you cannot take the natural logarithm of zero or a negative number, the domain of the base function \( y = \ln x \) would normally be \( x > 0 \). However, with the absolute value transformation, any negative x-values get reflected into positive x-values, making them suitable for the natural logarithm operation. Therefore, the domain of \( f(x) = \ln |x| \) is \( x \neq 0 \).

Step 3: Graph the Function

Sketch the base function \( y = \ln x \) for \( x > 0 \). This should be a curve increasing towards the right, starting from negative infinity at \( x = 1 \). To account for the absolute value transformation, reflect this graph about the y-axis to cover \( x < 0 \). There should be a vertical asymptote at \( x = 0 \).

Key Concepts

Domain of a Function

Understanding the domain of a function is like setting the ground rules for the problems you'll solve. The domain refers to all possible input values (x-values) that a function can accept. You can think of it as the list of x-values you can plug into the function without breaking any mathematical rules.

When working with logarithmic functions, such as the natural logarithm, you need to be careful with the inputs. The natural logarithm, denoted as \( \ln(x) \), only accepts positive numbers. This is because you cannot take the logarithm of zero or negative numbers. Thus, normally the domain would be \( x > 0 \).

In our specific function \( f(x) = \ln |x| \), we add the absolute value function. This allows both positive and negative x-values to be accepted, since applying an absolute value makes negative numbers positive. The target x-values exclude zero because the logarithm of zero is undefined. Therefore, the domain of \( f(x) = \ln |x| \) is \( x eq 0 \), meaning every real number except zero.

Natural Logarithm

The natural logarithm function, denoted as \( \ln(x) \), is a fundamental mathematics concept that gives the power to which the base \( e \) (approximately 2.718) must be raised to produce a given number. It is commonly used in both pure and applied mathematics.

Here are some basic properties of the natural logarithm:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln(e) = 1 \) because \( e^1 = e \).
• It is only defined for positive numbers; thus, it typically has a domain of \( x > 0 \).
• It is an increasing function, which means as x-values increase, \( \ln(x) \) also increases.

In transforming the natural logarithm with absolute value as in \( \ln |x| \), we broaden the scope to include both positive and negative values, while respecting the very nature and limits set by the \,\ln(\cdot) function itself. This tweaking of the function by using an absolute value allows for a complete picture across all real numbers except zero.

Absolute Value Function

The absolute value function is an important mathematical operation represented by two vertical bars around a number or variable, like \( |x| \). This function measures the magnitude of a number, disregarding its sign. Essentially, it reflects negative numbers onto the positive side, maintaining the distinction for positive numbers and zero.

Some characteristics of the absolute value function include:

• For any positive number \( x \), \(|x| = x\).
• For any negative number \( x \), \(|x| = -x\), turning it positive.
• Zero stays zero, so \(|0| = 0\).
• Graphically, it forms a v-shape that touches the origin \((0,0)\).

By introducing the absolute value function into our equation \( \ln |x| \), we enable the function to handle negative inputs by treating them as their positive counterparts. This action is visually represented by reflecting the graph of the natural logarithm, which is only defined for positive values, over the y-axis, thereby covering all the real numbers except zero. This reflection creates symmetry in the graph across the y-axis, enriching our understanding of the function.