Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 0.5 & \\ \hline 0.9 & \\ \hline 0.99 & \\ \hline 0.999 & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.5 & \\ \hline 1.1 & \\ \hline 1.01 & \\ \hline 1.001 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$f(x)=\frac{1}{x-1}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\frac{1}{x-1}\) is all real numbers except \(x = 1\). As \(x\) approaches 1, the function \(f(x)\) tends toward negative infinity from the left, and positive infinity from the right, confirming the existence of a vertical asymptote at \(x = 1\).

Step 1: Find the Domain

The domain of a function is the set of all possible input values (often \('x'\) values). For the function \(f(x)=\frac{1}{x-1}\), the denominator cannot be equal to 0, otherwise the function would be undefined. So, we set \(x - 1 = 0\), and solve for \(x\). From this, we see that \(x\) cannot be 1. Therefore, the domain of the function \(f\) is all real numbers except 1.

Step 2: Complete the Tables

To complete the tables for different \(x\) values, we substitute each \(x\) value into the function \(f(x)=\frac{1}{x-1}\). This will give the corresponding \(f(x)\) values. For example, to find the \(f(x)\) value when \(x = 0.5\), we substitute 0.5 into the function, giving us \(f(0.5) = \frac{1}{0.5 - 1} = -2\). We perform the same procedure for each \(x\) value, populating the tables with the corresponding \(f(x)\) values.

Step 3: Discuss The Behavior of f(x)

The function \(f(x)=\frac{1}{x-1}\) experiences a vertical asymptote at \(x = 1\). As \(x\) approaches 1 from the left, the denominator becomes increasingly smaller leading to a large negative value for \(f(x)\). As \(x\) approaches 1 from the right, the denominator instead becomes increasingly small positive, hence driving \(f(x)\) to large positive values. This can be confirmed from the values populated in the tables.

Key Concepts

Asymptote Behavior

Understanding asymptote behavior is essential when studying the characteristics of a function. An asymptote is a line that a graph of a function approaches but never actually touches. In the context of the function f(x) = \( \frac{1}{x-1} \), we observe a vertical asymptote at x = 1. A vertical asymptote represents a value that x may approach but cannot equal since the function would not yield a real number at that point.

As x gets infinitely close to 1 from the left side (x < 1), f(x) decreases without bound, meaning it will tend towards negative infinity. Conversely, as x approaches 1 from the right side (x > 1), f(x) increases without bound, or tends toward positive infinity. This dramatic change in function values when nearing an asymptote illustrates why it's a critical concept in analyzing a function's behavior.

Function Behavior Analysis

Analyzing the behavior of a function across different value ranges provides insight into its overall characteristics and how it behaves in various contexts. For the given function f(x) = \( \frac{1}{x-1} \), we use both calculus concepts and the function's nature to predict behavior at different points.

Aside from recognizing the asymptotic behavior near x = 1, it's also beneficial to explore how f(x) behaves at extreme values of x, such as when x is very large or very small. In our function, as x grows larger, the expression \( \frac{1}{x-1} \) approaches zero, indicating that the function flattens out and the graph gets closer and closer to the x-axis (x-axis acts as a horizontal asymptote).

Similarly, if x is a large negative number, f(x) will also approach zero. Through this analysis, we uncover that the function has two distinct behaviors, divided by the vertical asymptote at x = 1.

Table Completion with Function Values

Completing a table with function values is a practical way to evaluate how a function behaves for specific inputs and to visualize its patterns and trends. When inputting x values that are closer and closer to the asymptote such as 0.5, 0.9, 0.99, and so forth, the corresponding f(x) values will reveal the nature of the function near the excluded x value – in this case, 1.

To complete the provided tables, we calculate f(x) by substituting each x value into f(x) = \( \frac{1}{x-1} \). As we do so, we notice that values of x less than 1 result in negative f(x) values, which become more negative as x gets closer to 1. For values of x more than 1, f(x) becomes positive and increases as x approaches 1 from the right. This table filling exercise solidifies our understanding of the function's behavior near its vertical asymptote and reinforces the concept of limits and continuous change in the function's output.