Problem 266 - Calculus Volume 1

Question

Sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. \(\quad f(x)=\ln (-x)\)

Step-by-Step Solution

Answer

Domain: \((-\infty, 0)\); Range: \((-\infty, \infty)\); Vertical Asymptote: \(x = 0\).

Step 1: Identify the Transformation

The given function is \( f(x) = \ln(-x) \). This function is a transformation of the basic natural logarithm function \( f(x) = \ln(x) \). The transformation involves reflecting the graph of \( \ln(x) \) across the y-axis because of the negative sign inside the logarithm, resulting in \( \ln(-x) \).

Step 2: Determine the Domain

The natural logarithm function \( \ln(x) \) is defined for \( x > 0 \). Since we have \( \ln(-x) \), we need \( -x > 0 \), which implies \( x < 0 \). Therefore, the domain of \( f(x) = \ln(-x) \) is all negative real numbers: \( (-\infty, 0) \).

Step 3: Determine the Range

The range of the natural logarithm function \( \ln(x) \) is \((-\infty, \infty)\). The transformation into \( \ln(-x) \) does not affect the range, so it remains \( (-\infty, \infty) \).

Step 4: Find the Vertical Asymptote

As we approach the edge of the domain from the left, specifically as \( x \) approaches 0 from the left, \( \ln(-x) \) approaches \( \infty \). Thus, there is a vertical asymptote at \( x = 0 \).

Step 5: Sketch the Graph

Start by sketching the graph of the natural logarithm \( \ln(x) \), then reflect it across the y-axis to represent \( \ln(-x) \). Remember that the graph will dive down to the negative y-values far left and towards positive y-values near \( x = 0 \) from the left. Include the vertical asymptote at \( x = 0 \).

Key Concepts

Understanding the Domain and Range of Logarithmic Functions

The domain of a logarithmic function refers to the set of input values (or x-values) for which the function is defined. For our function, \( f(x) = \ln(-x) \), the key transformation is the negative sign inside the logarithm. Instead of dealing with positive x-values, we now need negative x-values, because the logarithm of a negative number is undefined. By setting \(-x > 0\), we solve for \(x < 0\). Thus, for this transformed function, the domain is all negative real numbers, \((-\infty, 0)\). This means the graph only exists on the left side of the y-axis.

The range of any basic logarithmic function, including \(\ln(x)\) or \(\ln(-x)\), is all real numbers, \((-\infty, \infty)\). This implies that no matter how you transform the graph, you can achieve any height on the y-axis by adjusting x-values within the domain. For \( f(x) = \ln(-x) \), every possible y-value remains accessible, making its range \((-\infty, \infty)\).

This combination of a negative domain with an all-encompassing range gives the graph its unique shape. Specifically, it will extend infinitely downward on the left and upward as it gets closer to the y-axis.

Exploring Vertical Asymptotes

Vertical asymptotes are lines that the graph approaches but never actually touches or crosses. For logarithmic functions like \( f(x) = \ln(-x) \), identifying the vertical asymptote is crucial in understanding its behavior near specific x-values.

In this function, observe that as x approaches 0 from the left (i.e., moving close to the y-axis but staying negative), the function value \( f(x) = \ln(-x) \) rises dramatically towards positive infinity, \( \infty \). This indicates a vertical asymptote at \( x = 0 \).

Vertical asymptotes reflect a boundary of the function’s domain. With \( \ln(x) \), the asymptote is typically at \( x = 0 \), but since our function uses \(-x\), it mirrors the behavior across the y-axis. Therefore, the graph will never meet or cross the line \( x = 0 \), reinforcing the fact that the function can never take x-values of 0 or more.

Logarithmic Function Graph Transformations

Graph transformations allow us to manipulate the appearance of a function’s graph in the coordinate system. For the function \( f(x) = \ln(-x) \), the transformation involved is a reflection across the y-axis.

To visualize this, start with the basic natural logarithm graph, \( \ln(x) \), which typically increases gradually from \( x = 1 \) to larger x-values and approaches negative infinity as x approaches zero from positive values. When you reflect \( \ln(x) \) across the y-axis, it mirrors over the vertical axis and gives \( \ln(-x) \).

Reflecting across the y-axis pivots the entirety of the graph into the negative x-domain, creating a symmetry where previously defined positive x-values now flip to negative ones.

This transformation effectively shifts the known behavior of the logarithm function to new territories – physically altering the graph’s path without affecting its fundamental properties, like range. Consequently, graph transformations are a powerful way to modify function visuals while preserving essential characteristics.