Problem 73 - Precalculus

Question

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

Step-by-Step Solution

Answer

The domain of \( f(x) = \log(x - 1) \) is \( x > 1 \) and the range is \((-\infty, \infty)\).

Step 1: Understand the Basic Logarithmic Function

The basic logarithmic function is \( g(x) = \log(x) \). Its graph is a vertical translation of the natural logarithm. The domain of \( g(x) \) is \( x > 0 \) and the range is all real numbers.

Step 2: Analyze the Transformation

The given function is \( f(x) = \log(x - 1) \). This function represents a horizontal shift of the basic logarithmic function. Specifically, it is shifted 1 unit to the right, which means the graph of \( g(x) = \log(x) \) will start at \( x = 1 \).

Step 3: State the Domain

For \( f(x) = \log(x - 1) \), the expression inside the logarithm must be greater than 0. Therefore, the domain is \( x - 1 > 0 \), which simplifies to \( x > 1 \). Thus, the domain of \( f(x) \) is \( x > 1 \).

Step 4: State the Range

The range of the logarithmic function is all real numbers because, as \( x \) approaches 1 from the right, \( f(x) = \log(x - 1) \) goes to negative infinity, and as \( x \) grows larger, it goes to positive infinity. Therefore, the range of \( f(x) \) is \( (-\infty, \infty) \).

Step 5: Graph the Function

Begin by plotting the basic log function \( y = \log(x) \). Then shift the entire graph one unit to the right to reflect the transformation \( f(x) = \log(x - 1) \). The vertical asymptote will be at \( x = 1 \). Plot a few points to confirm the shape, such as \((2, 0)\) and \((10, 1)\), because when \( x = 2 \), \( f(x) = \log(1) = 0 \), and when \( x=10 \), \( f(x) \) is approximately \(1\).

Key Concepts

Domain and Range

When working with any logarithmic function, it is crucial to understand the concepts of domain and range. The **domain** of a function refers to all possible input values (or "x" values) for which the function is defined. For a logarithmic function like the one we've examined, the inside of the logarithm, known as the argument, must be greater than zero. Therefore, for the function \( f(x) = \log(x - 1) \), you solve \( x - 1 > 0 \), resulting in \( x > 1 \). This means that the domain of \( f \) is all numbers greater than 1.

Now let's consider the **range**. This includes all possible output values (or "y" values) the function can achieve. A distinctive property of logarithmic functions is that their range is always all real numbers. This is because, as \( x \) approaches the lowest boundary from the right (that is 1, for \( x > 1 \)), the function outputs negative infinity. Conversely, as \( x \) increases, the function outputs positive infinity. Therefore, the range of \( f(x) \) is \((-\infty, \infty)\). Such understanding of the domain and range helps in identifying the behavior of the function.

Transformation Techniques

Transformation techniques involve modifying a function in a way that shifts, stretches, compresses, or reflects its graph while maintaining its basic shape. For logarithmic functions, these transformations help in translating the basic function \( g(x) = \log(x) \) into a new function.

In the function \( f(x) = \log(x - 1) \), we observe a **horizontal translation**. This happens because the function subtracts 1 from the argument. A horizontal shift always moves the graph along the x-axis. Specifically, \( x - 1 \) results in a translation rightwards by 1 unit. Hence, each point on the graph of \( \log(x) \) moves 1 unit to the right, shifting the vertical asymptote from \( x = 0 \) to \( x = 1 \). Utilizing transformations properly allows one to quickly understand how a function will appear when graphed and aids in identifying key features of the graph.

Graphing Functions

Graphing a logarithmic function involves plotting points and observing the shape and direction of the graph. For the function \( f(x) = \log(x - 1) \), you begin with the parent graph of \( g(x) = \log(x) \). After understanding the transformation, implement the horizontal shift of 1 unit to the right.

When graphing, it’s helpful to plot specific points to guide the drawing. Start with calculating a few values: like \( (2, 0) \), where \( f(x) = \log(1) = 0 \), and \( (10, 1) \), where \( f(x) \approx 1 \) for \( x = 10 \). These anchor points give clarity to the shape of the graph.

• The graph passes through these calculated points.
• This graph rises to the right and falls sharply to the left but never crosses the vertical asymptote at \( x=1 \).
• Always remember, the vertical asymptote is a line the graph approaches but never touches or crosses.

By visualizing the graph, individuals can better comprehend the nature of the logarithmic function, its growth, and any limitations imposed by its domain and range.