Problem 26 - College Algebra

Question

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{2 x}{x-6} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x=6\), horizontal asymptote at \(y=2\).

Step 1: Local Behaviour - Vertical Asymptote

The function \( f(x) = \frac{2x}{x-6} \) has a vertical asymptote where the denominator is zero. Solve \( x - 6 = 0 \) to find \( x = 6 \). Thus, as \( x \) approaches 6, \( f(x) \) tends to move towards positive or negative infinity depending on the direction of the limit.

Step 2: Horizontal Asymptote

To find the horizontal asymptote, consider the end behavior of \( f(x) \). Since the degrees of the numerator and the denominator are equal (both degree 1), the horizontal asymptote is given by the ratio of the coefficients of the highest degree terms. This gives \( y = \frac{2}{1} = 2 \). Thus, as \( x \to \pm \infty \), \( f(x) \to 2 \).

Step 3: Summary of Local and End Behavior

For the local behavior, \( f(x) \) has a vertical asymptote at \( x = 6 \). As \( x \to 6^+ \), \( f(x) \to +\infty \), and as \( x \to 6^- \), \( f(x) \to -\infty \). For end behavior, \( f(x) \to 2 \) as \( x \to \pm \infty \).

Key Concepts

Local Behavior

Local behavior in rational functions focuses on what happens near certain points of the function, particularly where the function may be undefined or behaves unusually. For the function \( f(x) = \frac{2x}{x-6} \), local behavior is mainly influenced by its vertical asymptote. A vertical asymptote occurs where the denominator equals zero, as this is where the function can become undefined. Here, you solve \( x - 6 = 0 \) to find that \( x = 6 \) is the location of the vertical asymptote.

As \( x \) approaches 6, the function \( f(x) \) will tend to go towards infinity or negative infinity. This is because as \( x \) gets very close to 6, the denominator approaches 0, causing the function's value to skyrocket in either the positive or negative direction. Which direction it goes depends on whether \( x \) approaches 6 from the left (\( x \to 6^- \)) or from the right (\( x \to 6^+ \)). These observations show how unexpectedly dramatic changes in behavior can occur in rational functions close to their vertical asymptotes.

End Behavior

End behavior of a rational function describes how the function behaves as the input \( x \) grows very large or very small (either positive or negative infinity). For the function \( f(x) = \frac{2x}{x-6} \), observing the end behavior helps us determine the horizontal asymptote.

End behavior rules imply looking at the ratio of the leading coefficients when the degrees of the numerator and denominator are the same. Here, both the numerator (\( 2x \)) and the denominator (\( x-6 \)) are of degree 1. The leading coefficients are 2 for the numerator and 1 for the denominator. Therefore, the function's horizontal asymptote is \( y = \frac{2}{1} = 2 \).

In simpler terms, as \( x \) goes to positive or negative infinity (\( x \to \pm \infty \)), the value of \( f(x) \) approaches 2. This behavior shows how rational functions "level off" at certain values as they 'move' to the far ends of the x-axis.

Vertical Asymptotes

Vertical asymptotes are critical parts of a rational function’s local behavior. They represent lines where the graph of the function heads towards infinity and thus are vertical lines on the graph at certain x-values. For \( f(x) = \frac{2x}{x-6} \), the vertical asymptote is at \( x = 6 \). To find it, look for values that make the denominator equal to zero, disrupting the function's definition.

When approaching this x-value from either direction:

• As \( x \to 6^- \), \( f(x) \to -\infty \)
• As \( x \to 6^+ \), \( f(x) \to +\infty \)

This indicates that as you get closer to \( x = 6 \), the function's values deviate widely, illustrating the "blowing up" effect around vertical asymptotes. Such asymptotes are vital for graphing and understanding areas where the function is not defined. They serve as warnings for these unbounded behaviors.

Horizontal Asymptotes

Horizontal asymptotes explore a function's end behavior, showing the value a function levels off to as \( x \) becomes very large or very small. For the rational function \( f(x) = \frac{2x}{x-6} \), you can determine the horizontal asymptote by examining the highest power terms. Since both the numerator (2x) and the denominator (x-6) have the degree of 1, horizontal asymptote computation involves the leading coefficients.

The formula \( y = \frac{2}{1} = 2 \) reveals that \( y = 2 \) is the horizontal asymptote. This tells us that as \( x \to \pm \infty \), the value of \( f(x) \) trends towards 2, indicating that the graph of the function will get close to the line \( y = 2 \) but never truly reaches or crosses it.

• Horizontal asymptotes tell us about eventual behavior, essential for predicting and understanding the "settling" point of rational functions in graphing.

These concepts are invaluable because they show rational functions' boundary behaviors, offering insights into their long-term trends.