Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the partial fraction decomposition for the rational function. Identify the graph of the rational function and the graph of each term of its decomposition. State any relationship between the vertical asymptotes of the rational function and the vertical asymptotes of the terms of the decomposition. $$y=\frac{2(4 x-3)}{x^{2}-9}$$

Step-by-Step Solution

Answer

The partial fraction decomposition for the given rational function is \(y = \frac{2}{x-3} - \frac{2}{x+3}\). The vertical asymptotes of the original function and each term of the decomposition are at x = 3 and x = -3

Step 1: Decompose the rational function into partial fractions

The decomposition of a rational function into partial fractions involves expressing it as a sum of simpler fractions. The denominator of the function \(y=\frac{2(4 x-3)}{x^{2}-9}\) can be factored as \((x - 3)(x + 3)\). Hence, the function can be decomposed into the form \(\frac{A}{x-3} + \frac{B}{x+3}\). The next step is to find the values for constants A and B.

Step 2: Determine the constants for the partial fractions

Setting the decomposed function equivalent to the original, \(\frac{2(4 x-3)}{x^{2}-9} = \frac{A}{x-3} + \frac{B}{x+3}\). To find A, let's set x = 3: this yields A = 2. Similarly, setting x = -3, this results in B = -2. Hence, the decomposed function is \(y = \frac{2}{x-3} - \frac{2}{x+3}\).

Step 3: Determine the vertical asymptotes of the original function and its decomposition

The vertical asymptotes of the function happen at values of x for which the function is undefined. Typically, these are the values that set the denominator to zero. From the original function, \(x^{2}-9 = 0\) when x = 3 or x = -3. Thus, these are the vertical asymptotes of \(y=\frac{2(4 x-3)}{x^{2}-9}\). Each term in the decomposition \(y = \frac{2}{x-3} - \frac{2}{x+3}\) also features these vertical asymptotes, indicating the roots of both denominators (x-3 and x+3) are at x = 3 and x = -3 respectively.

Key Concepts

Rational Functions

Rational functions are mathematical expressions where one polynomial is divided by another. The general form can be written as \(\frac{p(x)}{q(x)}\), where \(p(x)\) is the numerator polynomial and \(q(x)\) is the denominator polynomial. These functions are essential in algebra, calculus, and various applications in science and engineering.

An important aspect of rational functions is their behavior in relation to their denominators. When a denominator equals zero, it isn't defined, leading to discontinuities in the graph called vertical asymptotes. This is significant because it restricts the domains of these functions to exclude such points.

Understanding how to work with rational functions is pivotal. One method is partial fraction decomposition, which simplifies complex rational expressions into a sum of simpler fractions, as demonstrated in the textbook exercise solving \(y=\frac{2(4 x-3)}{x^{2}-9}\). The goal is to facilitate integration, differentiation, and graphing of the rational function.

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never touches or crosses. They signify points where a function heads towards infinity or negative infinity. Key in identifying vertical asymptotes is setting the denominator of a rational function to zero and solving for x.

For the function \(y=\frac{2(4 x-3)}{x^{2}-9}\), we determine vertical asymptotes at \(x=3\) and \(x=-3\) because these make the denominator, and hence the function, undefined. However, when analyzing the partial fraction decomposition into \(\frac{2}{x-3} - \frac{2}{x+3}\), we find the same vertical asymptotes. This correspondence confirms the link between the vertical asymptotes of the original function and each term in the decomposition, as they both originate from the denominator's factors.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler components, called factors, that when multiplied together, give back the original polynomial. Factoring is vital for simplifying expressions, solving equations, and in steps for partial fraction decomposition.

In our exercise, the polynomial \(x^{2}-9\) factored into \(x - 3)(x + 3)\). This step is critical to the decomposition process as it reveals the roots and potential vertical asymptotes of the rational function. It also sets the stage for the remaining work needed to simplify and illustrate the function.

Remember, comfortable factoring skills enhance one's ability to handle rational functions and their properties!