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Chapter 6: Problem 45

For the following exercises, sketch the graph of the indicated function. $$f(x)=\ln (-x)$$

Short Answer

Expert verified
The graph of \( f(x) = \ln(-x) \) is a logarithmic curve on \((-\infty, 0)\), decreasing as it moves left.

Step by step solution

01

Identify the Domain

The function given is \( f(x) = \ln(-x) \). For the natural logarithm function \( \ln(u) \) to be defined, its argument \( u \) must be positive. Thus, \( -x > 0 \), which implies \( x < 0 \). Therefore, the domain of the function is \((-\infty, 0)\).
02

Determine the Key Characteristics

Since \( f(x) = \ln(-x) \) is a transformed version of the natural logarithm function \( \ln(x) \), this function is defined and continuous on its domain \((-\infty, 0)\). As \( x \to 0^- \), \( f(x) \to -\infty \), making the y-axis (\( x = 0 \)) a vertical asymptote.
03

Sketch the Basic Shape

The graph of \( f(x) = \ln(-x) \) is a reflection of the graph of \( \ln(x) \) across the y-axis. It will have a general logarithmic curve shape, decreasing from \( x = 0 \) to the left towards negative infinity on x-axis.
04

Plot Points to Guide Curvature

Calculating a few key values can help inform the shape. For example, at \( x = -1 \), \( f(-1) = \ln(1) = 0 \). As \( x \) becomes more negative, \( f(x) \) increases; for example, \( x = -e \) gives \( f(-e) = \ln(e) = 1 \).
05

Draw the Graph

Using the information gathered, the graph should approach the vertical asymptote at \( x = 0 \) from the left and should show that \( f(x) \) increases without bound as \( x \to -\infty \). Plot the identified points and connect them smoothly, reflecting the general 'log' shape.

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is a special logarithm based on the constant \( e \), approximately equal to 2.71828. This function is widely used in exponential growth calculations and is known for its unique properties. One key feature is that the natural logarithm of \( e^x \) is \( x \), making it the inverse of the exponential function \( e^x \).
  • The natural logarithm is undefined for non-positive arguments; it only works with positive numbers.
  • It is a monotonically increasing function, meaning it consistently grows without decreasing as its input increases.
  • Its graph passes through the point \( (1,0) \), since \( \ln(1) = 0 \).
Understanding these aspects helps grasp how transformations affect the natural logarithm graph, especially for negative domain inputs like in \( f(x) = \ln(-x) \).
Function Domain
The domain of a function is the set of all input values (\( x \) values) for which the function is defined. For the function \( f(x) = \ln(-x) \), determining the domain requires considering where the natural logarithm is defined.
  • Since \( \ln(u) \) is only defined when \( u > 0 \), \( \ln(-x) \) exists only if \( -x > 0 \).
  • Solving \( -x > 0 \) leads to \( x < 0 \), restricting possible \( x \) values to any negative real number.
As such, the domain for \( f(x) = \ln(-x) \) is \( (-\infty, 0) \). This consideration ensures that the transformation involving \( -x \) shifts the usual domain of \( \ln(x) \) to the negative side of the axis.
Vertical Asymptote
In mathematics, a vertical asymptote of a function is a line \( x = a \) where the function values increase or decrease without bound as \( x \) approaches \( a \). For \( f(x) = \ln(-x) \), a vertical asymptote occurs at \( x = 0 \).
  • As \( x \) approaches 0 from the left (or negative side), the function value \( f(x) \) decreases towards negative infinity.
  • The graph of \( f(x) = \ln(-x) \) approaches but never touches the line \( x = 0 \).
This behavior mirrors the asymptotic nature of the natural logarithm \( \ln(x) \) in its positive form, but flipped across the y-axis for this transformed version.
Transformation of Functions
Transformation of functions involves shifting or altering the original function graph to create a new function form. For \( f(x) = \ln(-x) \), this represents a specific transformation of the natural logarithm \( \ln(x) \).
  • The transformation here is a reflection across the y-axis, switching positive \( x \) inputs to negative values, thus \( \ln(-x) \) reflects \( \ln(x) \).
  • This changes the usual rise of the logarithmic curve in the first quadrant to a decline in the fourth quadrant.
  • Key transformation points include ensuring the graph still maintains its characteristic shape, but inverted horizontally.
By understanding transformations, it becomes clear how \( \ln(-x) \) maintains its natural logarithm properties while adapting to a shifted domain and new orientation.

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