Problem 14 Sketch the graph of the function... [FREE SOLUTION]

Chapter 4: Problem 14

Sketch the graph of the function. $$ y=\ln |x| $$

Short Answer

Expert verified

The graph of the function $y=\ln|x|$ starts from negative infinity at $x=1$, rises slowly to the right, and is reflected over the y-axis, creating a mirror image for negative x-values with a vertical asymptote at $x=0$.

Step by step solution

Identify the function

The given function is $y=\ln|x|$, which is a modification of the traditional natural logarithm function. Because of the absolute value, the function will be defined for both positive and negative x-values.

Sketch the graph for positive x-values

For positive x-values, the function $y=\ln|x|$ would be the same as $y=\ln(x)$. The graph for this portion of the function would start at negative infinity when $x=1$, and then rise slowly to the right as x increases.

Reflect the graph for negative x-values

For negative x-values, the function $y=\ln|-x|$ would produce the same y-values as $y=\ln(x)$, because the absolute value makes the function symmetrical. This means we can reflect the right half of the graph over the y-axis to get the left half of the function, creating a vertical asymptote at $x=0$ as well.

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Key Concepts

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

Natural Logarithm Function

The natural logarithm function, commonly denoted as $ \ln(x) $, is a fundamental concept in calculus and is the inverse of the exponential function $ e^x $. Its domain consists of all positive real numbers, i.e., $ x > 0 $. It gives the power to which the base $ e $ must be raised to produce the number $ x $. This function is commonly used in various applications, such as calculating exponential growth or decay.

Important characteristics of the natural logarithm include:

Its graph passes through the point (1, 0), because $ \ln(1) = 0 $.
The function approaches negative infinity as $ x $ approaches zero from the right $ \lim_{x \to 0^+} \ln(x) = -\infty $.
It increases slowly without bound as $ x $ increases.

This function fundamentally requires positive $ x $-values, but in the context of $ \ln|x| $, it can handle both positive and negative values by making use of the absolute value function.

Absolute Value Function

The absolute value function, represented by $|x|$, is defined as the distance of a number $ x $ from zero on the number line, disregarding whether it is positive or negative. It can be mathematically described as:

$ |x| = x $ if $ x \geq 0 $
$ |x| = -x $ if $ x < 0 $

This function modifies $ ln(x) $ into $ ln|x| $, allowing the logarithmic function to extend its domain to negative values. Here's how it works for this graph:

For positive $ x $-values, $ |x| $ behaves just like $ x $, leaving the function as $ ln(x) $.
For negative $ x $-values, $ |x| $ turns $ x $ into $ -x $, still allowing the logarithm to proceed by treating the value as positive.

Therefore, the incorporation of the absolute value creates an effect of symmetry across the y-axis in the graph.

Graph Symmetry

Graph symmetry refers to the property of a graph looking identical on both sides of an axis, predominantly the y-axis or the origin. Symmetry simplifies graphing because it lets us understand the behavior of a function on one half and then mirror it without recalculating.

In the case of $ y=\ln|x| $:

For positive $ x $-values, it behaves like $ ln(x) $.
For negative $ x $-values, due to the absolute value, it mirrors $ ln(x) $ across the y-axis.

This particular symmetry is around the y-axis, making the graph appear the same on both the left and right sides. Thus, understanding one half of the graph suffices to envisage its entirety, thanks to its symmetry.

Vertical Asymptote

Vertical asymptotes occur in a graph where the function approaches infinity or negative infinity at certain x-values. They generally indicate values that are not in the function's domain.

For the function $ y=\ln|x| $, a vertical asymptote occurs at $ x=0 $. Here鈥檚 why:

The logarithmic function $ y=\ln(x) $ has an inherent vertical asymptote as $ x $ approaches zero from the right $ x \to 0^+ $.
Similarly, $ y=\ln|x| $ has this property at both sides of zero, since the absolute value function includes values coming from the left as $ x \to 0^- $.

This means the function approaches negative infinity as $ x $ approaches zero from both directions. The graph, thereby, develops this characteristic asymptotic behavior at $ x = 0 $, and never actually touches or crosses the y-axis.

91影视

Sketch the graph of the function. $$ y=\ln |x| $$

Short Answer

Step by step solution

Identify the function

Sketch the graph for positive x-values

Reflect the graph for negative x-values

Key Concepts

Natural Logarithm Function

Absolute Value Function

Graph Symmetry

Vertical Asymptote

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