Problem 66 - College Algebra with Corequisite Support

Question

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote. $$ f(x)=\frac{x}{x-3} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x=3\), horizontal asymptote at \(y=1\).

Step 1: Understand Vertical Asymptote

To find the vertical asymptote of a rational function, set the denominator equal to zero and solve for \(x\). For \(f(x)=\frac{x}{x-3}\), the denominator is \(x-3\). Setting it equal to zero gives \(x - 3 = 0\), so \(x = 3\) is the vertical asymptote.

Step 2: Analyze Function Behavior Near Vertical Asymptote

Create a table of values approaching \(x=3\) from both sides. Choose values such as \(x=2.9, 2.99, 3.1, 3.01\). \(f(x)\) approaches infinity as \(x\) approaches \(3\) from the left, and negative infinity as \(x\) approaches \(3\) from the right.

Step 3: Find Horizontal Asymptote

Consider the degrees of the polynomials in the numerator and denominator. For \(f(x)=\frac{x}{x-3}\), both degree are 1. The horizontal asymptote is \(y=\frac{1}{1}=1\).

Step 4: Examine Horizontal Asymptote Behavior

Create a table of values for large \(x\), such as \(x=10, 50, 100\). As \(x\) increases or decreases to very large values, \(f(x)\) approaches 1, showing \(y=1\) is the horizontal asymptote.

Step 5: Summarize Findings

The vertical asymptote occurs at \(x=3\), and near this point, \(f(x)\) tends to infinity on one side and negative infinity on the other. The horizontal asymptote is \(y=1\), approached by the function as \(x\) becomes very large or very small.

Key Concepts

Vertical Asymptote

In rational functions, a vertical asymptote occurs where the denominator is zero, leading to undefined values for the function. To find this asymptote for the function \( f(x) = \frac{x}{x-3} \), we set the denominator equal to zero: \( x - 3 = 0 \). Solving this equation gives us \( x = 3 \), which is the vertical asymptote.

As \( x \) approaches this point, the function value \( f(x) \) does not settle at a finite number but instead tends toward infinity or negative infinity. This behavior means:

• As \( x \) gets very close to 3 from values less than 3 (like 2.9, 2.99), \( f(x) \) becomes increasingly large.
• As \( x \) approaches 3 from values greater than 3 (such as 3.1, 3.01), \( f(x) \) reduces to very large negative values.

Understanding the behavior near vertical asymptotes helps predict where the function sharply rises or falls. This knowledge is crucial in graphing rational functions effectively.

Horizontal Asymptote

A horizontal asymptote indicates the value that a function approaches as \( x \) goes to positive or negative infinity. For the function \( f(x) = \frac{x}{x-3} \), we need to compare the degrees of the polynomials in the numerator and denominator.

Both polynomials here are of degree 1. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In this case, it is \( y = \frac{1}{1} = 1 \). This means:

• As \( x \) becomes larger and larger, the value of \( f(x) \) gets closer to 1.
• Similarly, as \( x \) becomes very small (in the negative direction), \( f(x) \) again approaches 1.

This asymptotic behavior suggests that at extreme values, the function trends towards a constant value, providing a horizontal line in the graph at \( y = 1 \). This helps in predicting the end behavior of the graph.

Function Behavior

Analyzing the behavior of a function involves understanding how it behaves around key points like asymptotes and at extreme values of \( x \). For \( f(x) = \frac{x}{x-3} \), we study both vertical and horizontal asymptotes to piece together a comprehensive picture.

Near the vertical asymptote at \( x = 3 \):

• As \( x \) approaches 3 from the left, \( f(x) \) tends towards infinity.
• Conversely, from the right of 3, \( f(x) \) drops towards negative infinity.

For horizontal asymptotic behavior around \( y = 1 \):

• For very large or very small \( x \), the function hovers around \( y = 1 \).

Function behavior considerations allow us to predict and sketch the graph accurately, indicating where lines approach but never touch asymptotes. This ensures a clear picture of how the function behaves across different values of \( x \).