Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-42: Sketch the graph of \(f\). $$ f(x)=\ln |x-1| $$

Step-by-Step Solution

Answer

The graph of \(f(x) = \ln |x-1|\) has two branches with a vertical asymptote at \(x = 1\), both approaching \(-\infty\) as they near \(x = 1\), and rising to \(\infty\) as \(|x|\) increases.

Step 1: Understanding the Function

The function given is \(f(x) = \ln |x-1|\), where \(\ln\) indicates the natural logarithm and \(|x-1|\) ensures we take the logarithm of a positive value since logarithm is only defined for positive values.

Step 2: Determining the Domain

The function \(|x-1|\) is positive and non-zero if \(x eq 1\). Thus, the domain of \(f(x) = \ln |x-1|\) is \(x \in (-\infty, 1) \cup (1, \infty)\). The function is not defined for \(x = 1\) because \(|x-1| = 0\) at \(x = 1\).

Step 3: Understanding the Behavior Near the Domain Boundaries

As \(x\) approaches 1 from the left (\(x \to 1^-\)), \(x - 1 \to 0^-\), thus \(f(x) = \ln |x-1|\to -\infty\). As \(x\) approaches 1 from the right (\(x \to 1^+\)), \(x - 1 \to 0^+\), so \(f(x) = \ln |x-1| \to -\infty\).

Step 4: Behavior as \(x\) Goes to Infinity or Negative Infinity

As \(x\) approaches infinity (\(x \to \infty\)), \(x - 1 \to \infty\), and \(f(x) = \ln |x-1| \to \infty\). Similarly, as \(x\) approaches negative infinity (\(x \to -\infty\)), \(x - 1 \to -\infty\) and \(f(x) = \ln |x-1| \to \infty\).

Step 5: Graphing the Function

The graph consists of two branches. For \(x < 1\), the plot approaches negative infinity as it nears \(x = 1\), and climbs up to positive infinity as \(x\) decreases further. For \(x > 1\), the situation is similar: the plot approaches negative infinity as it nears \(x = 1\), and climbs up to positive infinity as \(x\) increases further. Together, this creates a graph with a vertical asymptote at \(x = 1\), and each branch increases without bound away from the asymptote.

Key Concepts

Graphing Functions

Graphing a function involves understanding how the output, or y-value, changes as the input, or x-value, moves across the number line. Essentially, it’s a visual snapshot of all the ordered pairs \((x, f(x))\) that satisfy the function. For \(f(x) = \ln |x-1|\), this process requires special attention to the properties of logarithms and absolute values.

• The function \(\ln\) stands for the natural logarithm, defined only for positive numbers.
• The absolute value, \(|x-1|\), ensures that \(\ln\) only ever receives positive inputs.

This means we can encounter scenarios where the graph changes direction or style, such as breaks or asymptotes. To graph \(f\), plot a smooth curve starting from far left, approaching \(-\infty\) at \(x = 1\), then shoot upwards as \(x\) goes toward \(-\infty\). Similarly, do this for the far right side, but mirror the left in pattern.

Vertical Asymptote

A vertical asymptote is a place where a graph of a function approaches a specific x-value but never touches or crosses it. For our function, \(f(x) = \ln |x-1|\), we find a vertical asymptote at \(x = 1\). This occurs because as \(x\) gets closer to 1, the expression \(x-1\) gets closer to zero, causing \(\ln |x-1|\) to plummet towards negative infinity.
For function behaviors involving asymptotes:

• As \(x\) nears 1 from either the left or the right, \(f(x)\) decreases without bound.
• The function never achieves or crosses the x-coordinate of the asymptote.

Vertical asymptotes are critical in sketching graphs because they define the line near which functions rapidly change direction, adding accuracy and intuition to the graph's overall shape.

Domain of a Function

The domain of a function represents all possible input values (x-values) for which the function is defined. For \(f(x) = \ln |x-1|\), this means considering the condition that the argument of the logarithm must be positive and non-zero. The absolute value, \(|x-1|\), must be positive, implying that x cannot equal 1.
Thus, the domain becomes all real numbers except 1, expressed in interval notation as \((-\infty, 1) \cup (1, \infty)\). This format specifies:

• For x-values to the left of 1, including extremely negative numbers, the function can compute outputs successfully.
• For x-values directly at 1, the function is not defined, thus producing a gap or break in the domain.
• For x-values to the right of 1, even into positive infinity, the function remains defined and operational.

Understanding a function's domain is crucial, as it dictates where on the x-axis the function can legitimately be graphed.