Problem 40 - Calculus of a Single Variable

Question

Horizontal Asymptotes In Exercises \(39-42,\) 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 graph of the function \( f(x) = \frac{|3x + 2|}{x-2} \) has one horizontal asymptote, y=3, for x < -2/3, and none for x > -2/3.

Step 1: Define the regions based on the absolute term

Given the function \( f(x) = \frac{|3x + 2|}{x-2} \), we'll start by defining the regions based on where the term inside the absolute value is equal to zero. In other words, we solve for 3x + 2 = 0 which gives the solution \( x = -\frac{2}{3} \). Thus, we have two regions, one where x < -2/3 and the other where x > -2/3.

Step 2: Split and simplify the function

For the region where x < -2/3, we have \( f(x) = \frac{-3x - 2}{x-2} = \frac{3(x - 2)}{x - 2} - \frac{4}{x - 2} = 3 - \frac{4}{x - 2} \). For the region where x > -2/3, we have \( f(x) = \frac{3x + 2}{x-2} \).

Step 3: Identify horizontal asymptotes

The horizontal asymptote is the value that the function approaches as x tends towards infinity or negative infinity. Looking at our simplified function, we can see that as x tends to infinity or negative infinity, the term \(\frac{4}{x-2}\) would tend to 0. Hence, for x < -2/3, \( f(x) \) tends towards 3 and for x > -2/3, \( f(x) \) does not exist hence, there are no horizontal asymptotes in this range.

Key Concepts

Piecewise Functions

Piecewise functions are special types of functions that have different expressions based on the values of their input. They are like mathematical chameleons, changing their structure in different parts of their domain. For example, the function \( f(x) = \frac{|3x + 2|}{x - 2} \) is a type of piecewise function. In this case, the absolute value creates two distinct "pieces" based on the regions determined by whether the expression \( 3x + 2 \) is positive or negative.

• If \( x < -\frac{2}{3} \), the expression \( 3x + 2 \) is negative, and the function becomes \( \frac{-3x - 2}{x-2} \).
• If \( x > -\frac{2}{3} \), the expression \( 3x + 2 \) is positive, and the function simplifies to \( \frac{3x + 2}{x-2} \).

Understanding piecewise functions involves identifying these regions and applying the appropriate subsection of the function in each range. This capability allows us to see how a single function can behave differently within different parts of its domain.

Absolute Values

Absolute values may seem tricky, but they're just a way to ensure that whatever is inside the absolute value bars is non-negative. The absolute value of a number is its distance from zero on the number line, regardless of direction.
Let's consider the absolute value expression in our function: \( |3x + 2| \). This expression stays positive in all situations, altering its terms depending on whether \( 3x + 2 \) is positive or negative, leading to two pieces:

• When \( 3x + 2 > 0 \), the expression remains \( 3x + 2 \)
• When \( 3x + 2 < 0 \), it becomes \( -(3x + 2) \)

Thus, absolute values in this case create two regions for the function, highlighting their importance in piecewise delineation. They ensure that every outcome is positive and help in creating a clear framework for function partitioning.

Graphing Utilities

Graphing utilities, such as graphing calculators and software, are invaluable tools for visualizing functions and their behaviors. They can help you easily identify key characteristics like asymptotes, intercepts, and regions of increase or decrease. To explore the function \( f(x) = \frac{|3x + 2|}{x-2} \), a graphing utility can help:

• Display the curve of the function across different values of \( x \)
• Highlight where the function may have vertical or horizontal asymptotes
• Differentiate between the function's pieces in the graph, based on pieces defined by absolute conditions

These powerful tools allow students to connect the algebraic manipulation of piecewise-defined, absolute value-containing functions with a visual representation, deepening understanding and reinforcing learning through interactive exploration.

Limits at Infinity

When examining horizontal asymptotes, the concept of limits at infinity becomes particularly relevant. A horizontal asymptote describes the behavior of a function as \( x \) approaches infinity or negative infinity.
In our function \( f(x) = \frac{|3x + 2|}{x-2} \), we identify horizontal asymptotes by observing the limits. For instance:

• In the region \( x < -\frac{2}{3} \), we simplify the function to \( 3 - \frac{4}{x-2} \), and as \( x \) tends towards negative infinity, the term \( \frac{4}{x-2} \) tends to zero. Thus, the function approaches an asymptote at \( y = 3 \).
• In contrast, for \( x > -\frac{2}{3} \), there is no stable behavior as \( x \) tends to infinity, because the function becomes undefined. Here, no horizontal asymptote exists.

Understanding limits at infinity helps you predict and explain the end behavior of functions, critical in exploring the overall shape and direction of functions in mathematical analysis.