Question

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

Short 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-by-step solution

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 \(3x+2 < 0\).

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.

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.