Problem 37 - Precalculus with Calculus Previews

Question

In Problems 37 and 38 , sketch the graph of the given function \(f\). $$ f(x)=|\ln x| $$

Step-by-Step Solution

Answer

Graph \( y = \ln x \) for \( x \geq 1 \) and reflect \( y = -\ln x \) for \( 0 < x < 1 \). The domain is \( x > 0 \).

Step 1: Determine the Domain

The function is defined as \( f(x) = |\ln x| \). The natural logarithm \( \ln x \) is defined for \( x > 0 \). Thus, the domain of \( f(x) \) is all positive real numbers, \( x > 0 \).

Step 2: Identify Important Features of \( \ln x \)

The natural logarithm \( \ln x \) has some key features: it passes through the point (1,0), is undefined for \( x \leq 0 \), and increases to infinity as \( x \to \infty \). However, for values \( 0 < x < 1 \), the function value of \( \ln x \) is negative.

Step 3: Apply the Absolute Value Transformation

The absolute value \( |\ln x| \) affects the graph by reflecting any negative output values of \( \ln x \) into positive values. Specifically, for \( 0 < x < 1 \), \( |\ln x| = -\ln x \) and for \( x \geq 1 \), \( |\ln x| = \ln x \). This means the graph of \( f(x) \) is mirrored over the x-axis from the original negative \( \ln x \) values.

Step 4: Sketch the Graph of \( f(x) = |\ln x| \)

With these properties, we can now sketch the graph of \( f(x) = |\ln x| \). For \( x < 1 \), the graph is a reflection over the x-axis of the negative part of \( \ln x \). For \( x \geq 1 \), the graph of \( \ln x \) remains the same. The point (1,0) is on the graph, and the graph is always above the x-axis due to the absolute value.

Key Concepts

Absolute Value Transformation

Understanding the absolute value transformation is crucial when graphing functions like \( f(x) = |\ln x| \). The absolute value operation, represented by \( |x| \), essentially takes any negative number and converts it to a positive one while leaving positive numbers unchanged. In the context of functions, this means that any part of the graph that would dip below the x-axis is "flipped" above the x-axis.

For \( f(x) = |\ln x| \), this transformation affects values where the natural logarithm results in negatives. Specifically, \( \ln x \) gives negative values for \( 0 < x < 1 \) because the logarithm of numbers less than 1 is negative. The absolute value ensures that instead of dipping below the x-axis, these values appear as positive venues of \( -\ln x \).

Here’s a simple step-by-step recap:

• Negative outputs of \( \ln x \) (for \( 0 < x < 1 \)) are reflected above the x-axis, becoming positive.
• Positive outputs of \( \ln x \) (for \( x \geq 1 \)) remain unchanged.
• The graph of \( |\ln x| \) maintains the structure of \( \ln x \) for positive outputs and mirrors negative parts to be positive, always staying above the x-axis or on it at \( x=1 \).

Domain of Logarithmic Functions

The domain of a logarithmic function is a foundational element when you graph or manipulate the function. For \( f(x) = |\ln x| \), the key is to understand where \( \ln x \) is defined. The natural logarithm \( \ln(x) \) is only defined for positive numbers; hence, its domain is all real numbers greater than zero. This means for any function containing a natural logarithm, you can only input values \( x > 0 \).

Consider the following points for the domain of \( \ln(x) \):

• \( \ln(x) \) is undefined for \( x \leq 0 \).
• \( \ln(x) \) is continuous and defined for all \( x > 0 \).
• The domain of \( f(x) = |\ln x| \) thus also becomes \( x > 0 \).

Always remember, restricting the domain to positive values avoids any undefined scenarios and ensures the mathematical validity of the logarithmic function.

Natural Logarithm Properties

The natural logarithm \( \ln(x) \) has specific properties that affect its graph and applications. Recognizing these properties will help you understand transformations and domains better.

Key properties of \( \ln(x) \):

• Intercept: Passes through the point \( (1, 0) \), because \( \ln(1) = 0 \).
• Behavior Near Zero: As \( x \) approaches zero from the right (\( x \to 0^+ \)), \( \ln(x) \) tends to \( -\infty \).
• Growth: \( \ln(x) \) steadily increases as \( x \) increases, though its rate of increase slows.
• Undefined: For any \( x \leq 0 \), \( \ln(x) \) is undefined, reinforcing the importance of knowing the correct domain.

These properties illustrate why the function initializes from negative infinity as \( x \) is small (but positive), intersects the x-axis at \( x=1 \), and continues to rise slowly beyond. Understanding these will aid you in predicting how logarithmic functions behave and how transformations, such as absolute value, impact them.