Problem 84 - Precalculus

Question

Graph the logarithmic function using transformation techniques. State the domain and range of \(f\). $$f(x)=2 \ln (-x)$$

Step-by-Step Solution

Answer

Domain: \(x < 0\); Range: \((-\infty, \infty)\).

Step 1: Identify the Parent Function

The given function is a logarithmic function, and the parent function is the natural logarithm function, \( g(x) = \ln(x) \). This is the simplest form of the logarithmic function.

Step 2: Reflect the Function

The function \( f(x) = 2 \ln(-x) \) involves a reflection. The expression \( -x \) inside the natural logarithm reflects the graph of the parent function \( \ln(x) \) across the y-axis.

Step 3: Stretch the Function Vertically

The coefficient \( 2 \) in the function \( f(x) = 2 \ln(-x) \) indicates a vertical stretch by a factor of 2. This means each y-value of \( \ln(-x) \) is multiplied by 2.

Step 4: Analyze the Domain and Range

For the function \( f(x) = 2 \ln(-x) \), the domain is determined by \( -x > 0 \), which simplifies to \( x < 0 \). The range of any logarithmic function is \((-\infty, \infty)\).

Step 5: Graph the Transformed Function

Start with the graph of \( y = \ln(x) \). Reflect it over the y-axis to get \( y = \ln(-x) \). Then stretch it vertically by a factor of 2 to graph \( y = 2 \ln(-x) \).

Key Concepts

Transformation Techniques

Transformation techniques in logarithmic functions help us modify a graph's shape and position. For the function given, which is \( f(x) = 2 \ln(-x) \), we can identify several transformations from the base log function.

• Reflection: The presence of \( -x \) in the function \( \ln(-x) \) reflects the graph across the y-axis.
• Vertical Stretch: The coefficient "2" applied to \( \ln(-x) \) stretches the graph vertically. This means each y-coordinate is multiplied by 2.

By understanding these transformations, students can predict and accurately graph the function. It also links the algebraic manipulations to their geometric interpretations, creating a deeper understanding of function behavior.

Domain and Range

When dealing with logarithmic functions, recognizing the domain and range is crucial. The domain tells us what x values are permissible, while the range specifies the potential y values.

For \( f(x) = 2 \ln(-x) \), consider the implications of \( -x \):

• The input to the logarithm function, \( -x \), must be greater than 0 for the logarithm to exist. This condition \( -x > 0 \) implies that \( x < 0 \).
• Therefore, the domain of \( f(x) \) is all negative real numbers, \( x < 0 \).
• The range for any real logarithmic function, including this one, is \( (-\infty, \infty) \), as the outputs can stretch from negative to positive infinity.

Understanding these concepts allows students to determine where the graph exists on the x-axis and its possible y-values.

Vertical Stretch

A vertical stretch is one type of transformation that modifies the look of a graph along the y-axis. In our function, \( f(x) = 2\ln(-x) \), the multiplier "2" before the logarithm signifies that each y-value of its graph is twice as high compared to the graph of \( \ln(-x) \) alone.

• With a multiplier greater than 1, such as 2, the graph undergoes a stretch, pulling it away from the x-axis.
• Each point on the graph of \( \ln(-x) \) will be multiplied by 2, resulting in a steeper incline.
• This makes it visually apparent that the function grows quicker in terms of height with respect to the x-axis.

Recognizing a vertical stretch allows students to anticipate how quickly or slowly the function rises or falls.

Reflection Across the Y-Axis

In mathematics, reflections change the position of a graph significantly. For \( f(x) = 2 \ln(-x) \), reflecting across the y-axis is induced by the \(-x\) inside the logarithm function.

• This transformation flips the graph of the parent function \( \ln(x) \) to \( \ln(-x) \).
• Instead of the graph rising to the right, it will rise to the left as it is reflected horizontally over the y-axis.
• Combining reflection with other transformations such as a vertical stretch means students need to consider multiple shifts in graph positions.

Understanding graph reflections is essential, as it enables students to re-imagine the directions and tendencies of a function.