Problem 87 - Precalculus with Calculus Previews

Question

In Problems \(85-88\), discuss how the graph of the given function can be obtained from the graph of \(f(x)\) \(=\ln x\) by means of a rigid transformation (a shift or a reflection)? $$ y=\ln x^{-1} $$

Step-by-Step Solution

Answer

The graph of \( y = \ln x^{-1} \) is obtained by reflecting \( y = \ln x \) across the x-axis.

Step 1: Identify the Base Function

The base function provided is \( f(x) = \ln x \). This is a standard natural logarithm function graph passing through the point \((1,0)\) and continuously rising.

Step 2: Understand the Transformation Applied

The given equation is \( y = \ln x^{-1} \). This can be rewritten using the property of logarithms \( \ln a^b = b \ln a \) as \( y = - \ln x \). This indicates a transformation.

Step 3: Describe the Transformation

The transformation \( y = - \ln x \) represents a reflection of the graph of \( y = \ln x \) across the x-axis. Each point \((x, y)\) on \( y = \ln x \) becomes \((x, -y)\) on \( y = -\ln x \).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical functions that are the inverse of exponential functions. The natural logarithmic function, represented as \( f(x) = \ln x \), is a special type of logarithm where the base is the constant \( e \), approximately equal to 2.718. In the context of graph transformations, the graph of \( \ln x \) has a gentle curve that starts from negative infinity as \( x \) approaches zero from the positive side and rises without bound as \( x \) increases. This graph passes through the point \((1, 0)\) because \( \ln 1 = 0 \).
Understanding the properties of logarithmic graphs is crucial for transforming them. They exhibit vertical asymptotes along the y-axis as \( x \to 0^+ \). The domain of \( \ln x \) is \( x > 0 \), and the range is all real numbers \((-\infty, \infty)\). These properties are foundational when you start to apply transformations to these functions, helping you predict their behavior visually on a graph.

Rigid Transformations

Rigid transformations in mathematics involve shifting or reflecting a graph without altering its shape or size. These transformations include translations (shifts) and reflections. They help us manipulate the position of a graph without changing its essential appearance.

• **Translation (Shift):** This means moving the graph horizontally or vertically. For example, adding a constant \( c \) to \( f(x) = \ln x \) would move it upward or downward without changing its shape.


• **Reflection:** This flips the graph across a chosen axis. For example, if we consider \( f(x) = -\ln x \), it signifies a reflection across the x-axis.

Understanding these transformations allows us to apply them to different functions, manipulating their visual representation on a graph. In the given exercise, converting \( y = \ln(x^{-1}) \) to \( y = -\ln x \) exemplifies a reflection, a type of rigid transformation.

Reflection Across Axes

Reflection across axes refers to flipping the graph of a function over a specific axis, which in turn changes the sign of the function's values. In mathematical terms:

• **Reflection across the x-axis:** For a function \( y = f(x) \), this transformation becomes \( y = -f(x) \). Each point \((x, y)\) on the graph is transformed to \((x, -y)\).


• **Reflection across the y-axis:** For a function \( y = f(x) \), this transformation becomes \( y = f(-x) \). Here, each point \((x, y)\) becomes \((-x, y)\).

Applying these reflections changes how the graph is oriented in respect to the Cartesian plane. In our given exercise, transforming \( y = \ln(x^{-1}) \) into \( y = -\ln x \) involves reflecting the natural logarithm graph across the x-axis, effectively reversing the y-values while keeping the x-values constant.