Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. $$f(x)=\log _{3}(x-1)$$

Step-by-Step Solution

Answer

Graph a logarithmic function \( f(x) = \log_{3}(x-1) \) shifted right by 1 unit with an asymptote at \( x = 1 \). Plot points like \( (2, 0) \) and \( (4, 1) \) to guide the curve.

Step 1: Understand the Function

The function given is \( f(x) = \log_{3}(x-1) \). It is a logarithmic function with a base of 3. This function is a transformation of the basic logarithmic function \( \log_{3}(x) \), shifted right by 1 unit.

Step 2: Identify the Domain

Since the logarithm is defined only for positive values of its argument, we need \( x - 1 > 0 \). Thus, the domain of \( f(x) \) is \( x > 1 \).

Step 3: Determine Key Points

Choose key values of \( x \) to find specific points on the graph. Start with \( x = 2 \):\[ f(2) = \log_{3}(2 - 1) = \log_{3}(1) = 0 \]Then try \( x = 4 \):\[ f(4) = \log_{3}(4 - 1) = \log_{3}(3) = 1 \]

Step 4: Analyze the Asymptote

Logarithmic functions have vertical asymptotes where their arguments go to zero. With \( x - 1 \rightarrow 0 \), the asymptote for this function is at \( x = 1 \).

Step 5: Sketch the Graph

Plot the points identified in Step 3, such as \( (2, 0) \) and \( (4, 1) \). Draw a vertical asymptote at \( x = 1 \). The graph will be increasing, starting from the asymptote at \( x = 1 \) and moving through the plotted points.

Key Concepts

Domain of Logarithmic Functions

Understanding the domain of a logarithmic function is essential, as it defines all the possible input values we can use. For the function \( f(x) = \log_{3}(x-1) \), we must consider when the argument, which is \( x - 1 \), is positive. This means we set up the inequality \( x - 1 > 0 \). By solving this, we find that the domain is all values where \( x > 1 \).
This simply indicates that the function only exists for values greater than 1.
Below 1, the logarithmic function is not defined as logarithms of non-positive numbers are not real.

The domain plays a key role in determining the graph's behavior, as it tells us the portion of the x-axis over which the graph is drawn.
Logarithmic functions are continuous within their domain, increasing or decreasing without interruption.

Asymptotes

Vertical asymptotes are lines that a graph approaches but never touches or crosses. With the given function \( f(x) = \log_{3}(x-1) \), the vertical asymptote occurs at the point where the argument \( x - 1 \) becomes zero.
Since \( x - 1 = 0 \) solves to \( x = 1 \), the vertical asymptote is located at \( x = 1 \).

The presence of this asymptote has a meaningful impact on the graph's appearance. As the graph approaches \( x = 1 \) from the right, it sharply falls downward towards negative infinity without ever actually reaching \( x = 1 \).
Vertical asymptotes indicate a boundary within which the function is defined and outside of which it is not. Asymptotes help us visualize the limits and behavior of the function at the edges of its domain.

Transformations of Functions

Transformations alter the appearance and position of a function's graph without changing its basic shape. In the case of \( f(x) = \log_{3}(x-1) \), we recognize a transformation of the classic \( \log_{3}(x) \) function.
Here, the expression \( x - 1 \) indicates a horizontal shift.

Instead of \( \log_{3}(x) \) starting at the vertical asymptote of \( x = 0 \), shifting the function 1 unit to the right results in a new asymptote at \( x = 1 \).
This horizontal shift is a common transformation where the graph moves right if a positive number is subtracted from \( x \).

Graphing transformations require identifying actions like shifts, reflections, and stretches based on operations within the function. Visualization aids understanding of these transformations on the coordinate plane, as the graph traversal begins differently from the base function's origin.