Problem 35 - Precalculus with Calculus Previews

Question

Show that \(f(x)=\ln |x|\) is an even function. Rewrite \(f\) as a piecewise- defined function and sketch its graph. Find the \(x\) -intercepts and the vertical asymptote of the graph.

Step-by-Step Solution

Answer

The function \( f(x) = \ln |x| \) is even, with x-intercepts at \( x = 1 \) and \( x = -1 \), and a vertical asymptote at \( x = 0 \).

Step 1: Determine if the function is even

To determine if the function is even, we need to show that \( f(x) = f(-x) \). Let's compute \( f(-x) = \ln |-x| = \ln |x| = f(x) \). Since \( f(x) = f(-x) \) for all \( x eq 0 \), the function \( f(x) = \ln |x| \) is even.

Step 2: Rewrite as a piecewise function

The function \( f(x) = \ln |x| \) can be rewritten as a piecewise function to separate the positive and negative inputs:\[f(x) = \begin{cases} \ln(x), & x > 0 \ \ln(-x), & x < 0 \end{cases}\]This allows us to clearly see how the function behaves in different domains.

Step 3: Sketch the graph

To sketch the graph of \( f(x) = \ln |x| \), plot the curve \( y = \ln(x) \) for \( x > 0 \) and reflect it over the y-axis. The curve will approach the y-axis (\( x = 0 \)) but never touch it, suggesting a vertical asymptote there. For \( x < 0 \), the graph will mirror the positive side, forming a symmetric shape around the y-axis.

Step 4: Find the x-intercepts

The x-intercept of a function occurs when \( y = 0 \). For \( f(x) = \ln |x| \), this happens when \( \ln |x| = 0 \) which is true for \( |x| = 1 \). Thus, the x-intercepts are at \( x = 1 \) and \( x = -1 \).

Step 5: Identify the vertical asymptote

Since \( f(x) = \ln |x| \) approaches negative infinity as \( x \) approaches zero from either side (but is never defined at \( x = 0 \)), \( x = 0 \) is a vertical asymptote.

Key Concepts

Piecewise Function

A piecewise function is a function that's defined by different expressions depending on the input values. This helps to illustrate how a function behaves in individual segments of its domain. For the function \( f(x) = \ln |x| \), it makes sense to consider how the logarithm operates for both positive and negative values of \( x \).
To achieve this, we define the function piecewise:
\[ f(x) = \begin{cases} \ln(x), & x > 0 \ \ln(-x), & x < 0 \end{cases} \]
**Why Use a Piecewise Function?**

• **Clarity:** It simplifies the understanding of function behavior across different input ranges.
• **Graphing Assistance:** It aids in sketching graphs accurately by defining behavior around critical points.

Determining a piecewise function requires looking at how a function's expression needs to change based on the input's sign, commonly when absolute values are involved.

Vertical Asymptote

A vertical asymptote is a line that a curve appoaches but never actually meets. It represents the limits where a function grows towards infinity or decreases towards negative infinity. For the function \( f(x) = \ln |x| \), the vertical asymptote is found where the function is undefined or trends towards infinity.

**Identifying the Vertical Asymptote in \( f(x) = \ln |x| \):**

• The function is undefined at \( x = 0 \).
• As \( x \) approaches 0 from the positive or negative side, \( \ln(x) \) or \( \ln(-x) \) approaches negative infinity.

The graph of \( f(x) \) thus has a vertical asymptote at \( x = 0 \). This information helps us understand that as we move closer to \( x = 0 \), the values of the function will continue decreasing without bound, indicating a characteristic behavior of the function near this point.

x-intercepts

The \( x \)-intercept of a function is where the graph crosses the x-axis. This occurs when the output of the function, \( y \), equals zero. Figuring this out for our function \( f(x) = \ln |x| \) involves solving where \( \ln |x| = 0 \).
Setting \( \ln |x| = 0 \) gives us \(|x| = 1\) which implies:

• \( x = 1 \)
• \( x = -1 \)

This means the x-intercepts of the graph are located at these points. These intercepts tell us critical points where our function touches or crosses the x-axis, essential for understanding the overall shape of the graph. Recognizing x-intercepts can also aid in graphing and determining intervals where the function is positive or negative.