Problem 81 - Precalculus

Question

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

Step-by-Step Solution

Answer

The domain is \((-4, \infty)\) and the range is \((-\infty, \infty)\).

Step 1: Identify the Basic Function

The basic logarithmic function is given by \( f(x) = \ln(x) \), which has a domain of \( x > 0 \) and a range of all real numbers. It has a vertical asymptote at \( x = 0 \).

Step 2: Apply Horizontal Shift

The function \( f(x) = \ln(x+4) \) is a horizontal transformation of the function \( \ln(x) \). Since it is \( \ln(x+4) \), the graph is shifted 4 units to the left. This changes the vertical asymptote from \( x = 0 \) to \( x = -4 \).

Step 3: Determine the Domain

After considering the horizontal shift, the new domain is all \( x \) such that \( x+4 > 0 \). Solving for \( x \), we get \( x > -4 \). Hence, the domain of \( f(x) = \ln(x+4) \) is \((-4, \infty)\).

Step 4: Determine the Range

The range of logarithmic functions is always the set of all real numbers, \((-\infty, \infty)\), regardless of horizontal or vertical transformations.

Step 5: Graph the Function

Draw the shifted exponential curve which starts just to the right of the vertical asymptote \( x = -4 \) and increases slowly as \( x \) increases. Make sure to indicate the vertical asymptote at \( x = -4 \) on the graph.

Key Concepts

Transformations

Transformations in logarithmic functions can significantly alter the appearance of the graph. They help in repositioning the basic logarithmic graph to show various scales and shifts. For the function \( f(x) = \ln(x+4) \), the transformation applied is a **horizontal shift**. This specific transformation is common when a constant is added to the \( x \) variable inside the logarithm.

To visualize transformations, it's helpful to understand the effect of each type:

• **Translation**: Shifts the graph horizontally or vertically.
• **Scaling**: Alters the size of the graph along the axes.
• **Reflection**: Flips the graph over an axis.

In the case of a horizontal shift, the addition inside the logarithmic function, \(+4\), shifts the graph left by 4 units. Understanding transformations allows anyone to manipulate the graph of a logarithmic function and predict all changes it goes through.

Domain and Range

The **domain and range** of a function tell you where the function is defined and what possible output values it can have. When dealing with logarithmic functions, identifying the domain is crucial because the input values must make sense in the context of logarithms.

For \( f(x) = \ln(x+4) \), let's focus on the domain:

• Start with the condition for an **internal logarithm**: the input must be greater than zero.
• Here, we solve \( x+4 > 0 \), leading to \( x > -4 \).

So, the domain of \( f(x) \) is \((-4, \infty)\), meaning it accepts any real number greater than \(-4\).

The **range** of a logarithmic function like this one extends across all real numbers, \((-fty, \infty)\). This is because the output of a logarithm continues indefinitely in both the positive and negative directions as the input values change.

Horizontal Shift

Understanding the **horizontal shift** helps determine where a logarithmic graph begins and behaves. In the function \( f(x) = \ln(x+4) \), adding 4 inside the logarithm results in a leftward shift.

This type of shift doesn't just change the starting point but also affects where the graph has its vertical asymptote. Originally, the vertical asymptote for \( \ln(x) \) is at \( x = 0 \). However, due to the horizontal shift by 4 units left, it moves to \( x = -4 \).

Conclusively, recognizing horizontal shifts is critical when graphing as it may affect the graph's asymptotic behavior and the domain. By understanding how these transformations operate, one ensures accurate graphing and function interpretation.