Problem 44 - A Graphical Approach to Precalculus with Limits

Question

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=-\log _{2}(-x)$$

Step-by-Step Solution

Answer

Reflect the graph of \( g(x) = \log_{2}(x) \) over the y-axis, then over the x-axis.

Step 1: Understanding the Base Graph

The base function is \( g(x) = \log_{2}(x) \), which is defined for \( x > 0 \). The graph of \( g(x) \) is a logarithmic curve that passes through the point \( (1, 0) \) and approaches negative infinity as \( x \) approaches 0 from the right.

Step 2: Identifying Transformations in \( -\log_{2}(-x) \)

The function \( f(x) = -\log_{2}(-x) \) includes two transformations of the base graph \( g(x) \):1. \(-x\) indicates a reflection of the graph over the y-axis because it affects the input variable \( x \). This means the graph is reflected over the y-axis, flipping it horizontally.2. The negative sign outside the logarithm \(-\log_{2}\) applies a reflection over the x-axis to the graph, flipping it vertically.

Step 3: Determining the Domain

For the function \( f(x) = -\log_{2}(-x) \) to be defined, \(-x\) must be greater than 0. Therefore, the domain of \( f(x) \) is \( x < 0 \).

Step 4: Graphing the Transformed Function

To graph \( f(x) = -\log_{2}(-x) \), start with the base logarithmic graph restricted to \( x > 0 \). First, reflect it over the y-axis to get the graph of \( \log_{2}(-x) \) for \( x < 0 \). Then reflect this new graph over the x-axis to account for the negative sign outside, resulting in the graph of \( f(x) \), which will pass through \( (-1, 0) \) with vertical asymptote at \( x = 0 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are a key topic in understanding how graphs behave under various transformations. These functions are the inverses of exponential functions and are commonly expressed in the form \(y = \log_b(x)\), where \(b\) is the base of the logarithm. A common base you might encounter is base 2, giving the function \(g(x) = \log_2(x)\). The graph of a basic logarithmic function is a curve that increases slowly, passing through the point \( (1, 0) \) and stretching towards negative infinity as \( x \) approaches zero from the right.

Understanding logarithmic functions is essential because they are used to model many real-life phenomena, such as measuring sound intensity (decibels) or the pH in chemistry, which uses a log scale. These graphs typically feature a vertical asymptote along the line \(x = 0\), which the curve approaches but never touches.

Logarithmic functions are continuous and only defined for positive \( x \) values. It means any transformations involving negative inputs need special handling. Knowing these characteristics will help you navigate through different transformations and reflections of the graph.

Function Reflection

Function reflection involves flipping the graph of a function across either the x-axis or y-axis, dramatically changing its appearance and direction. In the context of the function \(f(x) = -\log_{2}(-x)\), two distinct reflections occur:

• First, the \( -x \) component reflects the graph over the y-axis. This means that the input values are inverted, switching the direction horizontally, as if you are flipping a pancake.
• Second, the negative sign outside the logarithm, \(-\log_{2}(x)\), reflects the graph over the x-axis, changing the output values' direction, inverting it vertically.

Through these reflections, the graph of the logarithmic function undergoes transformations drastically altering its shape. You may approach understanding reflection more effectively if you consider each axis separately, knowing that flipping changes only one aspect of the function—either the input or output—at any given time.
Employing reflection techniques provides insight into how graphs can be manipulated. They are a core element of graph transformations in algebra and calculus.

Domain of Functions

The domain of a function defines the set of input values \(x\) for which the function is defined. This concept is crucial when dealing with functions that have restrictions, like logarithmic functions. The original base function \(g(x) = \log_{2}(x)\) is defined only for \(x > 0\). This domain arises because you cannot take the logarithm of a non-positive number in real-number terms.

For the transformed function \(f(x) = -\log_{2}(-x)\), the domain restriction shifts due to the inner \(-x\). To satisfy \(-x > 0\), the values of \(x\) need to be less than zero, thus \(x < 0\). As you can see, simple transformations like modifying the input variable can change the allowed set of inputs for a function.

Understanding the domain helps ensure that you choose appropriate values to evaluate and graph the function effectively. Errors in determining the domain may lead to incorrect results or graphs, so always pay attention to transformations that impact the input variable.