Problem 63 - Precalculus

Question

Apply a graphing utility to graph \(y_{1}=\frac{2 x^{3}-8 x+16}{(x-2)^{2}\left(x^{2}+4\right)}\) and \(y_{2}=\frac{1}{x-2}+\frac{2}{(x-2)^{2}}+\frac{x+4}{x^{2}+4}\) in the same viewing rectangle. Is \(y_{2}\) the partial-fraction decomposition of \(y_{1} ?\)

Step-by-Step Solution

Answer

Yes, \\(y_2\\) is the partial fraction decomposition of \\(y_1\\).

Step 1: Simplify the Expressions

First, we note that both functions have common terms in their denominators: \(x-2\) and \(x^2+4\). For \(y_1\), rewrite the numerator \(2x^3-8x+16\) in terms of its factors and compare it with the summands on the right-hand side.

Step 2: Rewrite Function y1 Using Partial Fractions

To perform a partial-fraction decomposition, divide the terms of the numerator by the common factors in the denominator. Decompose the expression to find the structure that aligns with \(y_2\).

Step 3: Compare y2 with Partial Fraction Decomposition

Look at the partial fraction pattern:\( y_2 = \frac{1}{x-2} + \frac{2}{(x-2)^2} + \frac{x+4}{x^2+4} \).Compare these fractions with the decomposed terms of \(y_1\) to see if they match.

Step 4: Graph Both Functions

Use a graphing utility to plot the functions \(y_1\) and \(y_2\) over the same interval. Carefully set the domain to avoid the discontinuities at \(x=2\) and observe the behavior.

Step 5: Check Function Equivalence

Analyze if the graphs coincide everywhere else in the domain, meaning that \(y_2\) is an exact decomposition of \(y_1\) except at points where the functions are undefined.

Key Concepts

Graphing Utility

A graphing utility is a tool, often in the form of software or a calculator, that helps visualize mathematical functions. When dealing with complex functions, like rational functions involving partial fraction decomposition, a graphing utility allows us to observe the function's behavior over a set interval. To graph the functions \(y_1\) and \(y_2\), you need to:

• Enter the expressions for \(y_1\) and \(y_2\) into the graphing utility, making sure to input them accurately.
• Set a viewing window which covers an appropriate range for the \(x\)-values. This means choosing values just adequately large to get a clear picture, and avoiding regions where \(x = 2\), due to a vertical asymptote here.
• Analyze the produced graphs to identify similarities, differences, and potential asymptotic behavior.

While graphing, observe where the functions are undefined (like at \(x = 2\)). This helps you understand the discontinuities and how they affect the function's shape. Graphing not only confirms the partial fraction decomposition but also provides a visual check for equivalence.

Function Equivalence

Function equivalence means that two seemingly different expressions result in the same value for all points in their domain (excluding undefined points). After graphing \(y_1\) and \(y_2\), the main task is to determine if their graphs coincide across allowable values.

• As \(y_2\) represents the partial fraction decomposition of \(y_1\), there should be no distinction in behavior between the two graphs except around \(x=2\), where discontinuities occur.
• Check both sides of the discontinuity. If \(y_2\) is indeed an exact partial fraction decomposition, the graphs will look identical in offset areas.

If graphing alone is insufficient, examining algebraic forms might provide additional confirmation. This involves confirming that the decomposed version \(y_2\) rearranges to the form of \(y_1\) in terms of equivalent algebraic constituents. Identifying these equations as equivalent builds confidence in our decomposition and our initial algebraic operations.

Simplifying Expressions

Simplifying expressions involves rewriting functions in a form that is often easier to analyze or compare, such as partial fraction decomposition. This method is especially useful for integration or observing the structure of rational functions.

• Start by breaking down the numerator or finding common factors between the numerator and denominator.
• Use the pattern observed in the denominators \((x - 2)^2(x^2+4)\) to determine possible partial fraction terms.
• Attempt to rewrite \(y_1\)'s numerator, factoring when possible, to match it up with known components of \(y_2\).

When accomplished correctly, simplified expressions expose hidden relationships, making it easier to verify equivalence with a function like \(y_2\). In essence, you are looking at the simplest form to which an expression can be rewritten, aiding in visualization through tools like graphing utilities and fostering a deeper understanding of function behavior.