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Chapter 4: Problem 43

Exer. \(41-46\) : Sketch the graph of \(f\) $$f(x)=\ln |x|$$

Short Answer

Expert verified
The graph of \(f(x) = \ln |x|\) is symmetrical about the \(y\)-axis, with a vertical asymptote at \(x=0\). It extends from \(-\infty\) to \(-1\), then rises from negative infinity for \(x < 0\), and similarly for \(x > 0\).

Step by step solution

01

Understand the Definition of the Function

The function given is \(f(x) = \ln |x|\). This means we will use the natural logarithm of the absolute value of \(x\). The absolute value makes the function defined for all \(x eq 0\).
02

Determine the Domain of the Function

Since the logarithm is only defined for positive numbers, \(\ln |x|\) is defined for \(x > 0\) and \(x < 0\). The domain for this function is therefore \((-\infty, 0) \cup (0, +\infty)\).
03

Identify Key Points on the Graph

Select points to plot: \((-2, \ln 2), (-1, \ln 1), (1, \ln 1), (2, \ln 2)\). Note that \(\ln 1 = 0\) for both \(x = 1\) and \(x = -1\). These points help illustrate the symmetry of the graph about the \(y\)-axis.
04

Study the Behavior Near x = 0

As \(x\) approaches 0 from the positive side, \(\ln x\) approaches \(-\infty\), and from the negative side, \(\ln |x|\) also approaches \(-\infty\). The function has a vertical asymptote at \(x = 0\).
05

Plot the Graph Based on the Analysis

Draw the graph for \(x > 0\); this is the usual \(\ln x\) curve. For \(x < 0\), reflect the behavior of \(\ln x\) in the negative \(x\)-axis. The two branches of the graph are symmetric about the \(y\)-axis.

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Key Concepts

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

Natural Logarithm
A natural logarithm, often denoted as \(\ln x\), is a logarithm to the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It is widely used in mathematics due to its close relationship with growth rates and calculus.
  • The natural logarithm of a number tells you how many times you need to multiply \(e\) to get that number.
  • For instance, \( \ln e = 1 \) because \(e^1 = e\).
  • When you take the natural logarithm of a product or quotient, you can use properties such as \(\ln(ab) = \ln a + \ln b\) and \(\ln \left(\frac{a}{b}\right) = \ln a - \ln b\).
Understanding natural logarithms is key when working with exponential functions, as they are the inverse. This means if you have a function \(f(x) = e^x\), the inverse would be \(\ln x\).
It helps us understand how exponential growth or decay behaves over time.
Absolute Value
The absolute value of a number, represented as \(|x|\), is the non-negative value of \(x\) without regard to its sign. Absolute value measures the magnitude of a real number, basically how far it is from zero on the number line.
  • For a positive number like 3, \(|3| = 3\).
  • For a negative number like -3, \(|-3| = 3\).
  • If \(x = 0\), then \(|0| = 0\).
In the context of the function \(f(x) = \ln |x|\), the absolute value transforms any negative input into a positive one.
This adjustment broadens the applicability of the natural logarithm since it is only defined for positive values.
Domain and Range
In mathematics, the domain of a function refers to the set of all possible input values (\(x\)) the function can accept. For the function \(f(x) = \ln |x|\), we consider all possible \(x\) values for which the function is defined.
  • The domain for \(f(x) = \ln |x|\) is \((-\infty, 0) \cup (0, +\infty)\).
  • This is because the absolute value makes all non-zero numbers positive, enabling the logarithm operation.
  • \(x = 0\) is excluded as \(|0|=0\) and the natural logarithm is undefined for zero.
As for the range, it encompasses all possible outputs of the function.
  • Since \(\ln |x|\) produces every real number based on the input's absolute value, it means that the range is \(( -\infty, +\infty )\).
  • This is because as \(x\) gets closer to zero, \(\ln |x|\) trends towards negative infinity, and as \(|x|\) increases, the value rises.
Graphing Functions
Graphing functions is the process of drawing a function's curve on a coordinate plane. For \(f(x) = \ln |x|\), you would consider both positive and negative values of \(x\).
  • Start by plotting key points. You might choose points like \((1, \ln 1) = (1, 0)\) and \((-1, \ln 1) = (-1, 0)\).
  • Notice that when \(x\) is positive and close to zero, the graph goes downwards steeply, mimicking \(\ln x\).
  • For negative \(x\), it mirrors the positive side, creating symmetry about the \(y\)-axis.
  • The vertical asymptote at \(x = 0\) indicates that the graph never touches or crosses this line.
By plotting these points and observing the behavior approaching \(x = 0\), you reveal the function's structure which is critical in understanding its properties.
The key is recognizing the symmetry due to absolute value, leading to identical behavioron both sides of the \(y\)-axis.

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Most popular questions from this chapter

Bacterial growth The number of bacteria in a certain culture increased from 600 to 1800 between 7: 00 A.M. and 9: 00 A.M. Assuming growth is exponential, the number \(f(t)\) of bacteria \(t\) hours after 7: 00 A.M. is given by \(f(t)=600(3)^{t / 2}\) (a) Estimate the number of bacteria in the culture at 8: 00 A.M., 10: 00 A.M., and 11: 00 A.M. (b) Sketch the graph of \(f\) for \(0 \leq t \leq 4\)

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