Problem 64 - Precalculus with Limits

Question

HOW DO YOU SEE IT? Identify the graph of the rational function and the graph representing each partial fraction of its partial fraction decomposition. Then state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs representing the partial fractions of the decomposition. To print an enlarged copy of the graph, go to MathGraphs.com. $$\begin{aligned} \text { (a) } y &=\frac{x-12}{x(x-4)} \\\ &=\frac{3}{x}-\frac{2}{x-4} \end{aligned}$$ $$\begin{aligned} \text { (b) } y &=\frac{2(4 x-3)}{x^{2}-9} \\\ &=\frac{3}{x-3}+\frac{5}{x+3} \end{aligned}$$

Step-by-Step Solution

Answer

The vertical asymptotes of the original rational function and its partial fractions are identical. For equation (a), both the original function and its partial fractions have the asymptotes \(x = 0\) and \(x = 4\). Likewise, for equation (b), both the original function and its partial fractions have the asymptotes \(x = -3\) and \(x = 3\).

Step 1: Identify Rational Function Equation

For each question (a) and (b), identify the provided rational function; (a) \(y =\frac{x-12}{x(x-4)}\) and (b) \(y =\frac{2(4 x-3)}{x^{2}-9}\).

Step 2: Identify Partial Fractions Decomposition

Next, identify the decomposed forms of the rational functions; (a) transparently decomposes to \(y =\frac{3}{x}-\frac{2}{x-4}\) and (b) decomposes into \(y =\frac{3}{x-3}+\frac{5}{x+3}\).

Step 3: Identify the Vertical Asymptotes

Following the identification of the rational functions and their decompositions, the next step is to identify the vertical asymptotes. The vertical asymptotes of the rational functions are determined by setting the denominator of the function equal to zero. For equation (a), the vertical asymptotes obtained by setting \(x(x-4) = 0\) are \(x = 0\) and \(x = 4\). For equation (b), when \(x^{2}-9 = 0\), the vertical asymptotes are \(x = -3\) and \(x = 3\).

Step 4: Identify Relationship Between Asymptotes

The next step is to identify how the vertical asymptotes of the original functions relate to the asymptotes of the partial fractions. For equation (a), the partial fractions \(y =\frac{3}{x}\) and \(y =-\frac{2}{x-4}\) have vertical asymptotes at \(x = 0\) and \(x = 4\) respectively, identical to the original function. Meanwhile, for function (b), the partial fractions \(y =\frac{3}{x-3}\) and \(y =\frac{5}{x+3}\) have vertical asymptotes respectively at \(x = 3\) and \(x = -3\), again matching the original function. This suggests that the vertical asymptotes of the partial fractions make up the vertical asymptotes of the original rational function.

Key Concepts

Rational Functions

Rational functions are expressions that present as ratios of two polynomials.
These functions can be quite complex, but they follow a recognizable pattern.
You might find a rational function like \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
Key features of rational functions include:

• They can display diverse behaviors depending on the degrees of the polynomials involved.
  
• They are not defined at values of \( x \) that make the denominator zero, leading to vertical asymptotes or holes.
  

Partial fraction decomposition is a method frequently employed to rewrite a rational function as a sum of simpler fractions.
This transformation aids in easier analysis of the function’s properties, such as limits and asymptotes.

Vertical Asymptotes

Vertical asymptotes in mathematics symbolize lines that a graph approaches but never actually touches or crosses.
With rational functions, vertical asymptotes occur at inputs where the function is undefined, typically where the denominator equals zero.
To find these asymptotes for a rational function like \( \frac{x-12}{x(x-4)} \), you'd identify where \( x(x-4) = 0 \).
The solutions, \( x = 0 \) and \( x = 4 \), are the vertical asymptotes, indicating that the function will grow without bound as \( x \) approaches these values.
Some important points to consider:

• Vertical asymptotes reflect potential pitfalls in calculations or when graphing.
  
• The behavior of the function around these asymptotes can drastically affect the overall shape and interpretation of the graph.
  

Asymptotic Behavior

A function's asymptotic behavior describes how it behaves as the inputs approach certain limits.
For rational functions, this behavior provides a deeper understanding of the function’s growth and downfall near points of interest, like vertical asymptotes.
In particular, as \( x \) nears a vertical asymptote, the function’s values will tend to either infinity or negative infinity.
Observing this can help in predicting a graph's trend without fully sketching it.
The relationship between a function’s partial fraction decomposition and its asymptotic behavior is crucial for analysis:

• Comparing the asymptotic behavior of partial fractions and the original function helps in envisioning effects of each component on the whole function.
  
• Seeing the behavior near asymptotes can guide decisions in calculus problems, such as integration and limit finding.
  

Understanding this concept hones the visual and analytical skills needed for proficient function analysis.