Problem 38 - Essential Calculus

Question

In Exercises \(37-40\), use a graphing utility to graph the function and identify any horizontal asymptotes. $$ f(x)=\frac{|3 x+2|}{x-2} $$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{|3x+2|}{x-2}\) has two horizontal asymptotes, \(y=3\) and \(y=-3\), and a vertical asymptote at \(x=2\).

Step 1: Deal with absolute value

The absolute value can be defined as \(|x| = x\) for \(x \geq 0\) and \(|x| = -x\) for \(x < 0\). Thus the function can be divided into two parts: \(f_1(x) = \frac{3x+2}{x-2}\) for \(3x+2 \geq 0\) and \(f_2(x) = \frac{-3x-2}{x-2}\) for \(3x+2 < 0\).

Step 2: Find asymptotes

To find the horizontal asymptotes of the function, we examine the limit of the function as \(x \to \infty\) and \(x \to -\infty\). For \(f_1(x) = \frac{3x+2}{x-2}\), as \(x \to \infty\), \(f_1(x) \to 3\), and as \(x \to -\infty\), \(f_1(x) \to 3\). So \(y=3\) is a horizontal asymptote. As for \(f_2(x) = \frac{-3x-2}{x-2}\), as \(x \to \infty\), \(f_2(x) \to -3\), and as \(x \to -\infty\), \(f_2(x) \to -3\). So \(y=-3\) is a horizontal asymptote. Note that a vertical asymptote happens where the denominator of our function equals zero. So for \(x=2\) there's a vertical asymptote.

Step 3: Graph the function

Now, plot the function \(f_1(x) = \frac{3x+2}{x-2}\) for \(3x+2 \geq 0\) and \(f_2(x) = \frac{-3x-2}{x-2}\) for \(3x+2 < 0\) to double check the calculated values. A graphing utility such as a graphic calculator or a computer program like GeoGebra can be used for this task.

Key Concepts

Absolute Value Functions

An absolute value function involves expressions like \(|x|\), which measure the distance of a number from zero on the number line. It's always positive or zero, making it quite different from regular functions that can have negative outputs. Absolute value can be determined by:

• \(|x| = x\) for \(x \geq 0\)
• \(|x| = -x\) for \(x < 0\)

In the function \(f(x) = \frac{|3x+2|}{x-2}\), the absolute value makes the function behave differently depending on the sign of \(3x + 2\). This requires us to split the function into two parts: one for when \(3x + 2 \geq 0\) and another for when \(3x + 2 < 0\). Understanding the behavior of absolute value functions is crucial to analyze the full range of the given function.

Limits at Infinity

Limits at infinity are used to understand the behavior of a function as the input value grows very large positively or negatively. For rational functions like \(f(x) = \frac{3x+2}{x-2}\), we focus on the highest power of \(x\) in the numerator and denominator to determine these limits.

• As \(x \to \infty\), \(f(x) \sim \frac{3x}{x} = 3\)
• As \(x \to -\infty\), \(f(x) \sim \frac{-3x}{x} = -3\)

These simplified forms indicate that the behavior at these extremes remains constant, indicating the existence of horizontal asymptotes. Identifying these limits helps determine where the function levels out, which are important for graphing and understanding the long-term behavior of functions.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a function equals zero, causing the function to become undefined, often leading to a graph rising or falling sharply. For our function \(f(x) = \frac{|3x+2|}{x-2}\), the denominator \(x-2\) equals zero when \(x=2\). When this happens, neither \(f_1(x)\) nor \(f_2(x)\) is defined because you can't divide by zero. Thus, \(x=2\) is a vertical asymptote, and the graph will show a sharp discontinuity at this point, meaning the function rises or falls towards infinity as it approaches \(x=2\) from either side. Recognizing vertical asymptotes is vital for sketching the accurate graph of a function.

Graphing Rational Functions

Graphing rational functions like \(f(x) = \frac{|3x+2|}{x-2}\) requires an understanding of both horizontal and vertical asymptotes, along with considering absolute values that influence the graph's behavior. When graphing, it’s essential to:

• Identify any asymptotes: both horizontal \(y=3\) and \(y=-3\), and the vertical \(x=2\).
• Determine the intervals for absolute values and split the function into parts.
• Use a graphing utility to accurately draw the curve and visualize behavior near asymptotes.

Graphing utilities can simplify this job by allowing you to input complex functions and view their graphs on a screen. This can confirm analytical predictions and provide a precise visual understanding for often complex behaviors of rational functions.