Problem 61 - Precalculus

Question

Apply a graphing utility to graph \(y_{1}=\frac{x^{9}+8 x-1}{x^{5}\left(x^{2}+1\right)^{3}}\) and \(y_{2}=\frac{4}{x}-\frac{1}{x^{5}}+\frac{2}{x^{2}+1}-\frac{3 x+2}{\left(x^{2}+1\right)^{2}}\) in the same viewing rectangle. Is \(y_{2}\) the partial-fraction decomposition of \(y_{1} ?\)

Step-by-Step Solution

Answer

Graphing shows that \(y_2\) is the partial-fraction decomposition of \(y_1\) if graphs overlap completely.

Step 1: Understand the Functions

First, recognize that we are dealing with two distinct functions. One is given in a simplified, factored form (\[y_{1}=\frac{x^{9}+8x-1}{x^{5}(x^{2}+1)^{3}}\]), while the second one appears as a sum of individual terms that may hint at a partial fraction decomposition (\[y_{2}=\frac{4}{x}-\frac{1}{x^{5}}+\frac{2}{x^{2}+1}-\frac{3x+2}{(x^{2}+1)^{2}}\]). Our goal is to determine if the second function, \(y_2\), represents the partial-fraction decomposition of \(y_1\).

Step 2: Use a Graphing Utility

Use a graphing calculator or graphing software to plot both functions \(y_1\) and \(y_2\) in the same display window or viewing rectangle. Choose a domain where the important features of the graphs, such as intercepts, asymptotes, and behavior at positive and negative extremes, are visible. A typical domain could be \([-10, 10]\) for \(x\).

Step 3: Compare Graphs

Observe the two graphs. If \(y_2\) is indeed the partial-fraction decomposition of \(y_1\), the two graphs should overlap completely without any discrepancies. This means their values should match across the entire viewing window, showing that they are equivalent functions.

Step 4: Analyze Overlap and Features

Ensure that the two functions have identical vertical asymptotes, horizontal asymptotes, and intercepts. If \(y_2\) matches \(y_1\) completely, then \(y_2\) must correctly represent all these characteristics of \(y_1\).

Step 5: Conclusion from Graphs

If the graph confirms that the lines overlap perfectly, \(y_2\) is the partial-fraction decomposition of \(y_1\). Otherwise, \(y_2\) is not the partial-fraction decomposition.

Key Concepts

Rational Functions

Rational functions are mathematical expressions created by dividing two polynomial functions. The structure of a rational function is:

• The numerator is a polynomial.
• The denominator is a non-zero polynomial.

These functions can exhibit unique characteristics, including asymptotes and discontinuities, which arise from their denominators.
For example, the function \(y_1 = \frac{x^9 + 8x - 1}{x^5(x^2 + 1)^3}\) is a rational function because it is the ratio of a ninth-degree polynomial to an eighth-degree polynomial. When examining rational functions, consider factors like:

• The degree of the numerator versus the denominator, which affects horizontal asymptotes and end behavior.
• Common factors in the numerator and the denominator that may cancel out, possibly removing potential holes or simplifying the function.

Understanding rational functions help in breaking down more complex expressions, like partial fraction decompositions, to find simpler, equivalent expressions.

Graphing Utility

A graphing utility is a powerful tool often used to visualize complex mathematical functions. These utilities can include software applications, online tools, or graphing calculators.
Functions that might look complicated algebraically can be understood better when visualized.
A graphing utility provides an immediate visual representation that helps in:

• Identifying key features such as intercepts and asymptotes.
• Comparing multiple functions on the same axes to see similarities or differences.
• Understanding the behavior of functions over set intervals or domains, like \([-10, 10]\) for functions \(y_1\) and \(y_2\).

Graphing utilities allow for adjustments in viewing windows and scales, making them flexible in viewing specific features. When solving problems involving complex functions or exploring potential equivalences, like checking if \(y_2\) is the partial-fraction decomposition of \(y_1\), graphing utilities are invaluable to verify overlaps and predict behavior.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are crucial in studying rational functions as they indicate the behavior of the function beyond typical plotted points.

• Vertical Asymptotes: These occur at values of \(x\) which make the denominator zero. For example, the function \(y_1\) has vertical asymptotes at values that result from \(x^5(x^2 + 1)^3\) being zero.


• Horizontal or Oblique Asymptotes: Affects the function's behavior as \(x\) approaches infinity. They occur based on the relative degree of the polynomials in the numerator and denominator.

For \(y_1\), since the degree of the numerator (9) is higher than that of the denominator (8), it can have an oblique asymptote which will impact how \(y_1\) behaves as \(x\) becomes very large or very small.
Asymptotes are important in confirming the equivalence of two functions, like \(y_1\) and \(y_2\), by ensuring they share the same asymptotic behavior, contributing to the idea of partial-fraction decomposition when the characteristic behaviors match.