Problem 35 Show that $f(x)=\ln |x|$ is an... [FREE SOLUTION]

91影视

Precalculus with Calculus Previews

Dennis G. Zill, Jacqueline M. Dewar

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 35

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.

Short Answer

Expert verified

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 by step solution

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.

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.

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.

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 $.

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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.

Short Answer

Step by step solution

Determine if the function is even

Rewrite as a piecewise function

Sketch the graph

Find the x-intercepts

Identify the vertical asymptote

Key Concepts

Piecewise Function

Vertical Asymptote

x-intercepts

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