Problem 76 - Precalculus

Question

Graph the logarithmic function using transformation techniques. State the domain and range of \(f\). $$\ln (x-1)$$

Step-by-Step Solution

Answer

Domain: \( x > 1 \). Range: All real numbers.

Step 1: Identify the Basic Function

The basic logarithmic function is the natural logarithm, which can be expressed as \( f(x) = \ln(x) \). This function has a vertical asymptote at \( x = 0 \) and passes through the point \( (1,0) \). The domain is \( x > 0 \) and the range is all real numbers.

Step 2: Determine the Transformation

The given function is \( f(x) = \ln(x - 1) \). This is a horizontal shift of the basic logarithmic function to the right by 1 unit. This is because the transformation in the form \( f(x) = \ln(x - h) \) shifts the graph to the right if \( h > 0 \). Thus, our function shifts to the right by 1 unit.

Step 3: Apply the Transformation

With the horizontal shift to the right by 1 unit, the vertical asymptote of the function \( \ln(x - 1) \) moves from \( x = 0 \) to \( x = 1 \). The new graph passes through the point \( (2,0) \), formerly \( (1,0) \) on the basic graph.

Step 4: Determine the Domain and Range

The domain of \( f(x) = \ln(x - 1) \) is determined by the condition \( x - 1 > 0 \), which simplifies to \( x > 1 \). The range of this function remains the same as the basic natural logarithm function, which is all real numbers \( (-\infty, \infty) \).

Key Concepts

Graph Transformations

Graph transformations are techniques that help us shift, stretch, compress, or reflect graphs of functions without changing their basic shape. These transformations enable us to adapt a graph to represent different equations just by altering its position or size.
For logarithmic functions like the natural logarithm, transformations include:

• Shifting: Moving the graph horizontally or vertically.
• Stretching/Compressing: Changing the graph's width or height.
• Reflections: Flipping the graph across an axis.

By applying transformations, we can easily predict the new position of points and asymptotes, facilitating better graph analysis and understanding.

Domain and Range

Understanding the domain and range of functions is crucial for graphing and interpreting functions. The domain of a function represents all the possible input values (x-values) that the function can accept without resulting in undefined or imaginary outputs. In contrast, the range represents all the possible output values (y-values) the function can produce.
For the function \( f(x) = \ln(x-1) \):

• The domain is \( x > 1 \), meaning all x-values greater than 1.
• The range is all real numbers \((-\infty, \infty)\), implying the function can take any real number as a result.

This is because the function \( \ln(x-1) \) is a horizontal shift of the natural logarithm, restricted by x-values greater than 1.

Natural Logarithm

The natural logarithm is a fundamental mathematical function, denoted as \( \ln(x) \), which calculates the power to which e (approximately 2.718) must be raised to obtain a number x. It is commonly used in various fields, such as science, engineering, and economics, particularly when dealing with exponential growth or decay.
Key properties of the natural logarithm include:

• It has a vertical asymptote at \( x = 0 \).
• Its graph passes through \( (1, 0) \) because \( \ln(1) = 0 \).
• The domain is \( x > 0 \) while the range is all real numbers.

Functions like \( \ln(x-1) \) are transformations of this basic function, showcasing how changes to the equation affect the graph's characteristics.

Horizontal Shift

Horizontal shifts involve moving a function's graph left or right on a coordinate plane without altering its shape. This typically occurs in functions like logarithmic ones when there's a constant added or subtracted from the input variable (x).
The general form for a horizontal shift in logarithmic functions is \( \ln(x-h) \):

• If \( h > 0 \), the function shifts to the right by h units.
• If \( h < 0 \), it shifts to the left by h units.

For \( f(x) = \ln(x-1) \), the function moves to the right by 1 unit. This shift adjusts the vertical asymptote from \( x = 0 \) to \( x = 1 \), consequently altering the domain's starting point from \( x > 0 \) to \( x > 1 \). Understanding this transformation is vital because it changes where the function is defined and directly impacts the graph's visual representation.