Problem 72 - College Algebra Essentials

Question

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. CAN'T COPY THE GRAPH $$h(x)=\ln (-x)$$

Step-by-Step Solution

Answer

The graph of function \(h(x) = \ln (-x)\) is a reflection of \(f(x) = \log_{2} x\) through the y-axis. The domain of \(h(x)\) is \(x<0\) and its range is all real numbers. Its vertical asymptote is at \(x=0\).

Step 1: Graph the Base Function

Begin by sketching the base function \(f(x) = \log_{2} x\). This is a simple logarithm function that crosses the x-axis at \(x=1\) and has a vertical asymptote at \(x=0\).

Step 2: Transformation

The function \(h(x) = \ln (-x)\) actually represents two transformations of \(f(x) = \log_{2} x\): a reflection in the y-axis (as the \(x\) is negative) and a base change from 2 to \(e\) for the logarithm (as it is denoted by \(\ln\) instead of \(\log\)). However, as the change of base doesn't affect the base shape of the graph, the focus will be on the reflection. So, reflect the graph of \(f(x)\) over the y-axis to obtain the graph of \(h(x)\).

Step 3: Identify the Vertical Asymptote, Domain and Range

After reflection, the vertical asymptote of \(h(x) = \ln (-x)\) is at \(x=0\), just as with \(f(x) = \log_{2} x\). However, the domain and range change due to the reflection. The domain of \(h(x)\) is \(x<0\), as all negative values of \(x\) are valid for \(\ln (-x)\). The range stays the same as with \(f(x)\): \(-\infty , \infty\) (all real numbers).

Key Concepts

Vertical Asymptote

A vertical asymptote is a line that a graph approaches as it gets closer to a certain value of \(x\), but never actually touches or crosses that line. For logarithmic functions like \(f(x) = \log_{2} x\), the vertical asymptote occurs at \(x=0\). This happens because logarithms are undefined for zero and negative numbers.

When we apply a transformation to create the function \(h(x) = \ln(-x)\), the vertical asymptote still remains at \(x=0\). This is due to the reflection over the y-axis which doesn't change the location of the asymptote. It's important to remember that no matter the transformations applied, the fundamental logarithmic nature dictates where the graph approaches but never touches on the x-axis.

Domain of Logarithmic Functions

The domain of a logarithmic function consists of all the x-values for which the function is defined. For the base function \(f(x) = \log_{2} x\), the domain is \(x > 0\).

The transformation to \(h(x) = \ln(-x)\) involves reflecting the graph over the y-axis. This modifies the domain to \(x < 0\). In other words, \(h(x)\) is defined only for negative values of \(x\). This change in domain reflects the function's new orientation and helps identify inputs that are valid. Understanding how transformations affect the domain helps in graphing and analyzing the function's behavior.

Range of Logarithmic Functions

The range of a function tells us all the possible output values (y-values). For the function \(f(x) = \log_{2} x\), the range is \(-\infty, \infty\).

When transforming to \(h(x) = \ln(-x)\), the range remains unchanged. This means that regardless of whether the graph is reflected or shifted, the set of all possible y-values for a standard logarithmic function will always include every real number. This constancy in range is a hallmark of the logarithmic function, making it predictable yet versatile across transformations.

Reflection Over Axes

Reflection over axes is a type of transformation that flips a graph across a specified axis.

For the function \(h(x) = \ln(-x)\), there is a reflection over the y-axis. This occurs because of the negative sign in front of the \(x\), affecting how the graph is oriented compared to its base form \(f(x) = \log_{2} x\).

This reflection changes the direction of the graph but preserves certain properties like the vertical asymptote and range. Understanding reflections helps distinguish how changes in the equation translate into visual shifts on the graph, making it easier to grasp complex transformations.