Problem 58 - College Algebra

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. $$ g(x)=-2 \log _{2} x $$

Step-by-Step Solution

Answer

The vertical asymptote is at \(x=0\). The domain for both functions is \(x > 0\), and their range is all real numbers. The transformation inverts and stretches the base graph by a factor of 2.

Step 1: Understanding the Base Graph

The first step is understanding the base graph \(y = log_{2}(x)\). This is a basic logarithmic function with base 2. The graph crosses the x-axis at \(x=1\) and gets infinitely close to, but never touches, the y-axis, creating a vertical asymptote at \(x=0\). The domain is \(x > 0\) and the range is all real numbers.

Step 2: Applying Transformations

Next, we consider the transformation effects on our base function to plot \(g(x)=-2 log_{2}(x)\). The multiplication of the logarithm by -2 inverts the graph over the x-axis and stretches it vertically by a factor of 2.

Step 3: Graphing and Features of Transformed Function

The transformed function \(g(x)\) also has a vertical asymptote at \(x=0\). However, unlike the base graph, \(g(x)\) is always below the x-axis due to the negative sign. The domain remains \(x > 0\) and the range is all real numbers because the graph will extend indefinitely both upwards and downwards.

Key Concepts

Graph Transformations

Understanding graph transformations is crucial when working with functions like logarithms. To start, we have a base function:

• Base Function: \(y = \log_{2}(x)\)

This is our "starting point" for any transformations. The exercise then gives us a transformation to apply:

• Transformed Function: \(g(x)=-2 \log_{2}(x)\)

To achieve this, let's break it down:

• First, the negative sign before the \(2\) indicates a reflection over the x-axis. This means that wherever the original graph goes above the x-axis, it will now reflect below it.
• Secondly, the "2" in front of the logarithm means that the graph will stretch away from the x-axis by a factor of 2. So, points on the graph will look "taller" or in this case, further away in the vertical direction from the x-axis.

These transformations help us adjust the graph's shape without changing its fundamental properties.

Vertical Asymptote

A vertical asymptote in a graph of a function is a vertical line that the graph approaches but never touches or crosses. This occurs in logarithmic functions like our base \(f(x) = \log_{2}(x)\), resulting in a vertical asymptote at the y-axis or at \(x = 0\).

In the function \(g(x) = -2 \log_{2}(x)\), the vertical asymptote remains unchanged. This means the function cannot accept any value of \(x\) which is less than or equal to zero.

The asymptote reflects the behavior that as \(x\) gets closer to zero, the value of \(f(x)\) dives away to negative infinity (in the original graph) or positive infinity in the case of the reflected function \(g(x)\). The concept of an asymptote is pivotal in understanding the graph's structure as it guarantees certain behavior near critical points.

Domain and Range

The domain and range of a function give us a sense of the possible input (x-values) and output (y-values) respectively.

For the base function \(f(x) = \log_{2}(x)\):

• Domain: \(x > 0\) because logarithms are undefined for non-positive x-values.
• Range: all real numbers \, \((-\infty, \infty)\) because a logarithmic function can output any real number as \(x\) increases.

When we apply transformations to get \(g(x) = -2 \log_{2}(x)\), these properties remain mostly the same:

• Domain: \(x > 0\), as transformations don’t change where the logarithm is defined.
• Range: all real numbers \, \((-\infty, \infty)\).

In essence, these properties determine the "allowable" and achievable values on our graphs.

Base 2 Logarithms

Base 2 logarithms are logs where the base number is 2. Logarithms answer the question "to what power must the base be raised, to produce a given number?"

The function \(f(x) = \log_{2}(x)\) specifically means we find how many times we multiply 2 to reach the number \(x\).

• For example, if \(x = 8\), then \(\log_{2}(8) = 3\) because \(2^3 = 8\).
• Another example, \(\log_{2}(1/2) = -1\) because \(2^{-1}=1/2\).

Base 2 logarithms are particularly useful in computing and digital applications because of their connection to binary systems. In our graphical context, understanding the base of the logarithm helps determine how quickly the function grows and how the graph "behaves." As \(x\) increases, the function output climbs more slowly, never reaching zero again once positive, defining the typical logarithmic curve.