Problem 43 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The function \(f(x)=\ln(-x)\) is defined for \(x < 0\), with a vertical asymptote at \(x = 0\), and resembles a flipped \(\ln(x)\) graph extending leftward.

Step 1 - Understand the Domain

The domain of a natural logarithm function is limited to positive values of the argument. Since we have \(-x\) inside the logarithm, this means that \(-x > 0\) or \(x < 0\). Thus, the domain of \( f(x) = \ln(-x) \) is all negative real numbers.

Step 2 - Identify Key Features

Since \( f(x) = \ln(-x) \) is similar to \( \ln(x) \) but applies to negative values of \(x\), it will have similar properties: - It will have a vertical asymptote at \(x = 0\).- As \(x\) approaches 0 from the left, the function will tend to \(-\infty\).- As \(x\) approaches \(-\infty\), the function will tend towards \(+\infty\).

Step 3 - Plot Points for Reference

Choose a few sample points to help sketch the graph:- At \(x = -1\), \(f(-1) = \ln(1) = 0\).- At \(x = -e\), \(f(-e) = \ln(e) = 1\).These points provide a guide for plotting.

Step 4 - Sketch the Graph

Using the key features identified and reference points, draw the graph:- Start from the vertical asymptote at \(x = 0\), with the curve moving from \(-\infty\).- Pass through the point \((x, y) = (-1, 0)\), moving steadily upwards.- Continue the pattern towards \(-\infty\), where it approaches \(+\infty\). Remember, the graph never crosses the vertical line \(x = 0\).

Key Concepts

Understanding the Function Domain

When dealing with natural logarithm functions, it is crucial to understand the domain. The natural logarithm, denoted as \( \ln(x) \), is typically defined for positive values of \( x \). However, our function is \( f(x) = \ln(-x) \), which involves \(-x\) instead of \(x\). This changes the domain significantly. For \( \ln(-x) \) to be valid, the argument \(-x\) must be positive. Thus, we have \(-x > 0\). By solving this inequality, we arrive at \( x < 0 \). This means the domain of \( f(x) = \ln(-x) \) consists of all negative real numbers. Whenever you work with logarithmic functions, always check the argument to determine the domain. In summary:- The function \( \ln(-x) \) is valid only when \(-x\) is positive.- Thus, the valid domain is all negative real numbers (\( x < 0 \)).- Make sure to verify these conditions whenever you're dealing with logarithms that involve negative signs inside.

Identifying the Vertical Asymptote

A vertical asymptote occurs when a function approaches infinity or negative infinity as it nears a certain value of \(x\). For the natural logarithm function, this is a typical feature. In our function \( f(x) = \ln(-x) \), we encounter a vertical asymptote when the function's argument approaches zero. Since we set \(-x > 0\), the asymptote appears as \( x \) approaches 0 from the left, but never actually reaches it. As this happens, \( f(x) \) shoots down to \(-\infty\). Here are some important points to remember:- The vertical asymptote for \( \ln(-x) \) happens at \( x = 0 \).- The graph will stretch vertically downward as \( x \) approaches 0 from the negative side.- The closer \( x \) gets to 0 (from the left), the more pronounced the decline towards negative infinity becomes.Thus, understanding the vertical asymptote is key to successfully sketching the logarithmic graph. Always consider where the function could head towards infinity or negative infinity.

Graph Sketching of the Logarithmic Function

Sketching the graph of a function like \( f(x) = \ln(-x) \) requires identifying key characteristics like intercepts, asymptotes, and overall behavior as \( x \) moves through its domain. Start by marking any reference points you know.Use these guidelines to effectively sketch the graph:- Notice the point \((x, y) = (-1, 0)\) where \( x = -1 \) gives \( f(x) = \ln(1) = 0 \).- Another key point is \((x, y) = (-e, 1)\) where \( f(-e) = \ln(e) = 1 \), providing a concrete place on the plot.From here, outline the pattern of the graph:- Begin near the vertical asymptote. As \( x \) approaches zero from the negative side, draw the ends of the curve plunging towards \(-\infty\).- Pass through the key points as you draw from the asymptote. This means, from just beyond \( x = 0 \), steadily progress through \((-1, 0)\) while moving upwards.- Continue extending the curve as \( x \) heads to negative infinity, guiding it towards \(+\infty\).Ultimately, the log function sketch will never cross \( x = 0 \), and it will handily curve from negative infinity through plotted points towards positive infinity. Be sure the graph reflects these dynamics accurately.