Problem 82 - Precalculus

Question

Graph the logarithmic function using transformation techniques. State the domain and range of \(f\). $$f(x)=\ln (4-x)$$

Step-by-Step Solution

Answer

Domain: \((-\infty, 4)\); Range: \((-\infty, \infty)\). Graph involves a right shift and reflection over the y-axis.

Step 1: Identify the Parent Function

The given function is based on the natural logarithm: \( f(x) = \ln (4-x) \). The parent function for this is \( g(x) = \ln(x) \), which is the natural logarithm function defined for \( x > 0 \).

Step 2: Identify Transformations

The function \( f(x) = \ln (4-x) \) can be rewritten as \( \ln(-(x-4)) \). This involves a horizontal shift of the logarithm graph 4 units to the right and a reflection over the y-axis.

Step 3: Determine the Domain

The expression inside the logarithm, \( 4-x \), must be greater than zero for the function to be defined. Thus, \( 4-x > 0 \) leads to \( x < 4 \). Therefore, the domain of \( f(x) \) is \( (-\infty, 4) \).

Step 4: Determine the Range

Since logarithmic functions can take on any real number as a result and transformations do not affect the range, the range of \( f(x) = \ln (4-x) \) is \( (-\infty, \infty) \).

Step 5: Graph the Function

Start with the parent function \( \ln(x) \). Shift the graph 4 units to the right to correspond to \( \ln(x-4) \). Finally, reflect this graph over the y-axis to represent the transformation \( \ln (4-x) \). The vertical asymptote will also shift to \( x = 4 \).

Key Concepts

Transformation Techniques

Logarithmic functions involve transformations that can shift, reflect, and even stretch or compress the graph from its basic parent function form. For the given function, \( f(x) = \ln (4-x) \), we start with the natural logarithm function \( g(x) = \ln(x) \), which is our parent function.

In transforming \( \ln(x) \) to \( \ln(4-x) \), we need to consider two main transformations:

• **Horizontal Shift**: Rewriting \( \ln(4-x) \) as \( \ln(-(x-4)) \), we see that it requires a right shift of 4 units, since we adjust the function to \( \ln(x-4) \).
• **Reflection**: The negative sign in \( \ln(-(x-4)) \) indicates a reflection of the graph over the y-axis.

These transformations help us visualize how the graph behaves and changes from its familiar shape, allowing us to graph the function accurately.

Domain and Range

Understanding the domain and range of a logarithmic function is crucial in knowing where the function can "live" and what values it can take. For \( f(x) = \ln(4-x) \), we need to ensure the expression inside the logarithm \( 4-x \) is positive, as logarithms of non-positive numbers are undefined.

To determine the domain:

• Set \( 4-x > 0 \) which simplifies to \( x < 4 \).
• Thus, the domain of \( f(x) \) is \((-\infty, 4)\), meaning the function is defined for all \( x \) less than 4.

As for the range:

• Logarithmic functions can output any real number, which means the range is \((-\infty, \infty)\).

Despite the horizontal shift and reflection, these transformations do not affect the range.

Graphing Logarithmic Functions

When graphing logarithmic functions, it's essential to start with the parent function and apply the transformations step by step. Begin with \( g(x) = \ln(x) \), which increases logarithmically and is undefined for \( x \leq 0 \).

For \( f(x) = \ln(4-x) \):

• **Start by shifting** the graph of \( \ln(x) \) four units to the right, resulting in a graph similar to \( \ln(x-4) \).
• **Then, reflect** the entire graph over the y-axis to adjust for the negative sign, achieving the correct form \( \ln(4-x) \).

The graph will trend downward as it approaches the vertical asymptote at \( x = 4 \), reflecting the fact that the expression \( 4-x \) approaches zero. This comprehensive visualization helps in understanding the shape and behavior of the logarithmic function.