Problem 58 - Algebra and Trigonometry

Question

In Exercises \(53-58,\) 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 for the function \(g(x)=-2 \log_{2}x\) is \(x=0\). The domain of the function is \((0,+\infty)\) and the range of the function is \(-\infty , +\infty\).

Step 1: Sketch the Graph of Basic Function

Start by plotting the basic graph of \(f(x)=\log _{2} x\). The graph of \(y=\log_bx\) for any base \(b>1\) intrinsically has a vertical asymptote at \(x=0\), thus for \(f(x)\), the vertical asymptote is \(x=0\). This is essentially a curve which starts from negative infinity at \(x=0\) and increases slowly as \(x\) increases.

Step 2: Apply the Transformations

To transform this graph to represent the function \(g(x)=-2 \log _{2} x\), apply the following transformations: 1. Reflect the graph of \(f(x)\) across the x-axis because of the minus sign in \(g(x)\). 2. Vertical stretch (expansion) by a factor of 2 due to presence of coefficient 2 in \(g(x)\). The vertical asymptote remains \(x=0\).

Step 3: Determine Domain and Range

From the graph of \(g(x)\), it is observed that the function exists for all positive \(x\), as log is defined only for \(x>0\). Thus, the domain of \(g(x)\) is \((0,+\infty)\). The transformed function continuously decreases as \(x\) increases, thus the range of \(g(x)\) is \(-\infty , +\infty\).

Key Concepts

Logarithm Transformations

When working with logarithmic functions like these, transformations are a necessary tool for graphing and understanding their behaviors. Logarithm transformations follow the general rules of function transformations, applied to the unique characteristics of logarithms.

For our example function, the base logarithm we start with is \(f(x) = \log_2 x\). One transformation we apply is a reflection over the x-axis, which is observed when the logarithmic function is multiplied by a negative number, in this case, \'-2\textquotesingle. This flips the graph upside down. Another important transformation is the vertical stretch by a factor of two. This is done due to the multiplier (the coefficient in front of the log function) affecting the 'steepness' of the graph. The greater the absolute value of the coefficient, the steeper the graph becomes at every point.

A transformation does not affect the location of the vertical asymptote; it remains at the same x-value as the original function's asymptote. In this problem, despite the negative sign and the stretch, the vertical asymptote for both \(f(x)\) and \(g(x)\) is at \(x=0\).

Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never touches or crosses. It represents a boundary beyond which the function cannot be defined or does not exist. With logarithmic functions like \(f(x) = \log_2 x\) and the transformed \(g(x) = -2 \log_2 x\), we encounter a vertical asymptote where the argument of the log (the 'x' in this case) equals zero.

For logarithmic functions, the vertical asymptote occurs at \(x=0\) because the logarithm of zero is undefined. Logarithms can only take positive values as their input, making the asymptote an essential characteristic of their graphs. When graphing \(g(x)\), even after applying transformations, this vertical asymptote at \(x=0\) remains in place as a silent sentinel, dictating the domain of the function and reflecting the inherent properties of logarithms.

Function Domain and Range

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined, while the range refers to the set of possible output values (y-values) that the function can produce.

In the case of our logarithmic function \(g(x) = -2 \log_2 x\), we identify the domain by looking at where the logarithm is defined. Since we can't take the log of zero or negative numbers, the domain is all positive numbers, written as \((0, +\infty)\). The range, on the other hand, is a result of the transformations. As the original log function increases indefinitely, the reflection due to the '-2' causes \(g(x)\) to decrease indefinitely. Therefore, the range of \(g(x)\) is the entire set of real numbers, indicated as \((-\infty, +\infty)\).

Understanding the domain and range of transformations is crucial, particularly with logarithms, because it informs us about the behavior of the function and the scope of its y-values for the corresponding domain. This understanding enables us to graph the function accurately and to anticipate its growth or decay.