Problem 114 - Precalculus

Question

$$\text { Graph the function } f(x)=\left\\{\begin{array}{ll}-\ln (-x) & x<0 \\\\-\ln (x) & x>0\end{array}\right.$$

Step-by-Step Solution

Answer

The function is not defined at \( x = 0 \) and has vertical asymptotes on both sides approaching \( x = 0 \).

Step 1: Understand the Function Definition

The function \( f(x) \) is defined piecewise for two intervals: \( x < 0 \) and \( x > 0 \). For \( x < 0 \), the function is \( f(x) = -\ln(-x) \). For \( x > 0 \), the function is \( f(x) = -\ln(x) \). The function is not defined for \( x = 0 \).

Step 2: Analyze the Behavior for \( x < 0 \)

For \( x < 0 \), we use the function \( f(x) = -\ln(-x) \). Note that \(-x \) will be a positive number since \( x \) is negative. The natural logarithm function is defined for positive inputs, so as \( x \) approaches 0 from the negative side, \( -x \) approaches 0 from the positive side, and \( -\ln(-x) \) will approach infinity. As \( x \) becomes more negative, \( -ln(-x) \) decreases towards negative infinity. Thus, there is a vertical asymptote at \( x = 0 \).

Step 3: Analyze the Behavior for \( x > 0 \)

For \( x > 0 \), the function is \( f(x) = -\ln(x) \). The natural logarithm of \( x \) approaches negative infinity as \( x \) approaches 0 from the positive side, causing \( -\ln(x) \) to approach positive infinity. As \( x \) increases, \( -\ln(x) \) becomes more negative, tending towards negative infinity. Hence, there is also a vertical asymptote at \( x = 0 \).

Step 4: Drawing the Graph

To graph the function, split the number line into two parts. For \( x < 0 \), sketch a curve that starts from negative infinity moving towards positive infinity as \( x \) approaches 0. For \( x > 0 \), start from positive infinity and move the curve towards negative infinity as \( x \) increases. Ensure there is a vertical asymptote at \( x = 0 \), and no point exists at \( x = 0 \) itself.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm with the base \( e \), where \( e \approx 2.71828 \). It is widely used in mathematics, primarily because of its unique properties and its relationship to calculus and exponential functions. The natural logarithm is only defined for positive values of \( x \).

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln(e) = 1 \) because the base is \( e \), meaning \( e^1 = e \).

When dealing with the natural logarithm, particularly in the context of piecewise functions as seen in the exercise, keep in mind:

• The logarithm will only work for positive real numbers. This is why \( -\ln(-x) \) makes sense only when \( x < 0 \) because \(-x \) becomes positive.
• Natural logarithms change values very slowly. It takes a significant change in \( x \) to produce a noticeable change in \( \ln(x) \).

Vertical Asymptote

A vertical asymptote is a line that a graph approaches but never actually touches or crosses. In the context of the function \( f(x) \) from the exercise, the key feature occurs at \( x = 0 \). The function is not defined at this point and approaches infinity or negative infinity as \( x \) gets closer to zero from both the left and right.For \( x < 0 \), as noted:- The graph approaches positive infinity as \( x \) nears 0, since \(-\ln(-x)\) increases indefinitely.For \( x > 0 \):- The graph approaches positive infinity as \( x \) nears 0 because \(-\ln(x)\) increases without bound.Vertical asymptotes are critical for understanding a graph's shape. They indicate where the graph's behavior changes dramatically and where a piecewise function such as \( f(x) \) is undefined.

Graphing Functions

Graphing functions, particularly piecewise functions, involves plotting different parts of an equation across specified intervals of \( x \). For the function in the exercise, each piece of the function exists in its domain (either \( x < 0 \) or \( x > 0 \)) and reacts differently.Here's a step-by-step approach to graph such functions:- **Identify domains**: Clearly define the intervals. In this case, \( x < 0 \) and \( x > 0 \), ensuring a vertical asymptote helps highlight the boundary at \( x = 0 \).- **Evaluate limits at boundaries**: Determine the behavior of the functions as \( x \) approaches certain critical points. For the graph, observe how the function behaves approaching the asymptote from both sides.Visual representation is key with piecewise functions. It shows how the function transitions from one behavior to another at the vertical asymptote. The graph will have two separate sections, highlighting the sharp contrast as the piecewise function transitions across its defined domains.