Problem 5 - Precalculus Mathematics for Calculus

Question

The following questions are about the rational function $$ r(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has vertical asymptotes \(x=\) __________ and $$x=$$ __________.

Step-by-Step Solution

Answer

Vertical asymptotes of \(r(x)\) are \(x = -2\) and \(x = 3\).

Step 1: Identifying Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is equal to zero, as long as the numerator is not also zero at the same points.

Step 2: Set Denominator to Zero

To find the values of \(x\) that cause the denominator to be zero, set \((x+2)(x-3) = 0\). This will give us two equations to solve.

Step 3: Solve for First Vertical Asymptote

Set \(x+2 = 0\) and solve for \(x\): \[x+2=0\] \[x=-2\] Therefore, \(x = -2\) is one vertical asymptote.

Step 4: Solve for Second Vertical Asymptote

Next, set \(x-3 = 0\) and solve for \(x\): \[x-3=0\] \[x=3\] Therefore, \(x = 3\) is the second vertical asymptote.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are crucial features of rational functions. They are vertical lines on a graph where the function approaches infinity or negative infinity. These occur at the values of \(x\) that make the denominator of the rational function zero, provided the numerator is not zero at the same points. Knowing where these vertical asymptotes are helps us understand the behavior of a function as it approaches these critical \(x\)-values.
To find vertical asymptotes, you will:

• Identify the rational function's denominator.
• Set the denominator equal to zero and solve for \(x\).
• Ensure the numerator does not equal zero at those \(x\) values.

The positions of vertical asymptotes tell us where the function is undefined. This concept is key to sketching a rational function's graph.

Denominator Zeros

The zeros of the denominator in a rational function are pivotal in determining vertical asymptotes. A rational function has the form \( r(x) = \frac{p(x)}{q(x)} \), where \(p(x)\) and \(q(x)\) are polynomial expressions. The zeros of \(q(x)\) are the \(x\) values that make \(q(x) = 0\).
To find these:

• Write down the denominator of the rational function, which is \((x+2)(x-3)\) in this case.
• Set each factor of the denominator to zero.
• Solve these simple equations for \(x\).

In our function, \((x+2)(x-3) = 0\) yields two solutions: \(x = -2\) and \(x = 3\). These \(x\) values show where the potential vertical asymptotes of the function occur. They are critical in analyzing the function's behavior around these points.

Solving Equations

Solving equations is an essential skill when working with rational functions to find vertical asymptotes. It involves setting expressions in the function equal to zero and finding the \(x\) values that satisfy these conditions. For our rational function \( r(x) = \frac{(x+1)(x-2)}{(x+2)(x-3)} \), we set the denominator \((x+2)(x-3)\) equal to zero.
To do this, break it down into simpler equations:

• For \(x+2=0\), subtract 2 from both sides to get \(x = -2\).
• For \(x-3=0\), add 3 to both sides to achieve \(x = 3\).

These steps identify the \(x\) values that make the denominator zero, thus indicating where vertical asymptotes might occur. Understanding how to solve these equations allows you to effectively deal with rational functions and anticipate their behavior.