Problem 53 - Algebra 2

Question

CHALLENGE Write three rational functions that have a vertical asymptote at \(x=3\) and a hole at \(x=-2 .\)

Step-by-Step Solution

Answer

Three possible functions: \(\frac{(x+2)}{(x+2)(x-3)}\), \(\frac{(x+2)(x-1)}{(x+2)(x-3)}\), \(\frac{2(x+2)(x+4)}{(x+2)(x-3)}\).

Step 1: Understanding the Problem

To create a rational function with a vertical asymptote at \(x=3\) and a hole at \(x=-2\), we need the denominator of our rational function to have factors corresponding to these points. A vertical asymptote occurs where the denominator is zero and the factor is not canceled out. A hole occurs where the numerator and denominator are zero at the same point and their common factor cancels out.

Step 2: Constructing the Denominator

For the vertical asymptote at \(x=3\), include \((x-3)\) in the denominator. For the hole at \(x=-2\), include \((x+2)\) in both the numerator and the denominator to allow it to cancel out.

Step 3: Construct Examples

1. Example 1: \(f(x) = \frac{(x+2)}{(x+2)(x-3)}\)2. Example 2: \(g(x) = \frac{(x+2)(x-1)}{(x+2)(x-3)}\)3. Example 3: \(h(x) = \frac{2(x+2)(x+4)}{(x+2)(x-3)}\)In each case, \((x+2)\) cancels out, creating a hole at \(x=-2\), and \((x-3)\) remains in the denominator, ensuring a vertical asymptote at \(x=3\).

Step 4: Verification

For each function, confirm that the vertical asymptote occurs at \(x=3\) because the \((x-3)\) term in the denominator does not cancel. Confirm that there is a hole at \(x=-2\) by ensuring that the \((x+2)\) term cancels in both the numerator and denominator.

Key Concepts

Vertical Asymptote

Vertical asymptotes are a fundamental aspect of rational functions and occur where the function, as written, becomes undefined by division by zero. If we have a rational function \[ f(x) = \frac{p(x)}{q(x)} \]where both \(p(x)\) and \(q(x)\) are polynomials, a vertical asymptote is present at any value of \(x\) where the denominator \(q(x)\) equals zero, and importantly, the numerator \(p(x)\) does not equal zero at that same point.For example, with a vertical asymptote at \(x=3\), the denominator could contain a factor of \((x-3)\). Therefore, as \(x\) approaches 3, the function grows towards infinity or negative infinity.

• A factor in the denominator like \((x-3)\) indicates where the function can "blow up" or become infinite.
• The existence of a vertical asymptote helps us understand the behavior of the function near specific points on the graph.

Hole in a Graph

A hole in the graph of a rational function occurs when there is an indeterminacy due to the same zero appearing in both the numerator and denominator, which causes a factor to cancel out. This type of point is often called a removable discontinuity. A hole at \(x=-2\) suggests that both the numerator and the denominator share a factor of \((x+2)\). When these common factors are canceled, it leaves behind a rational expression that doesn't explicitly show the hole, but the indeterminacy at that point remains.

• The factor \((x+2)\) in both the numerator and the denominator ensures a hole at \(x=-2\).
• The graph looks continuous and smooth elsewhere, but mathematically the function is undefined exactly at \(x=-2\).

Canceling the factor doesn’t remove the indeterminate nature at \(x=-2\), but it does simplify the function to one that doesn’t "see" the hole when graphed outright. Such subtleties in rational functions richly illustrate why it is crucial to factor and simplify correctly!

Numerator and Denominator

Rational functions involve both a numerator and a denominator, integral to its behavior and the graph's shape. Let's break down each part:The **numerator**, noted as \(p(x)\) in the expression\[ f(x) = \frac{p(x)}{q(x)} \]is pivotal in determining the overall value of the rational function at given points. If both the numerator and denominator equal zero at a given value of \(x\), it contributes to creating a hole there. On the other hand, specific zeros in the numerator influence the x-values where the function itself attains a value of zero.The **denominator**, \(q(x)\), should not be zero for the function to remain defined because division by zero is impossible in standard arithmetic. Factors found in the denominator are where you should examine for potential asymptotes or holes.

• A common factor between the numerator and the denominator indicates a possible hole.
• Denominators with factors not canceled by the numerator point towards vertical asymptotes.

Understanding the interplay between the numerator and denominator is crucial when identifying vertical asymptotes and holes, as these determine the distinct characteristics of rational function graphs.