Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to verify that the expressions are equivalent. Verify the results algebraically. $$y_{1}=\frac{x^{4}+x^{2}-1}{x^{2}+1}, \quad y_{2}=x^{2}-\frac{1}{x^{2}+1}$$

Step-by-Step Solution

Answer

The detailed solution to this problem requires the use of a graphing calculator or utility. The graphical representation would indicate whether the functions are equivalent or not. However, to be entirely sure, a mathematical verification is performed by simplifying both expressions and comparing them. The results will determine if the two expressions are indeed mathematically equivalent.

Step 1: Graph the Equations

Use a graphing utility to graph the two equations \(y_{1}=\frac{x^{4}+x^{2}-1}{x^{2}+1}\) and \(y_{2}=x^{2}-\frac{1}{x^{2}+1}\). Plot these equations on the same graph. Observe carefully to see if there are points where the two graphs intersect or overlap. The two functions are equivalent if their waves are identical.

Step 2: Observing the graphs

Keep in mind that if two graphs are the exact mirror of each other along the x-axis, then the functions they represent are equivalent. Analyze the graphs to see if the curves coincide. If they match visually, it would suggest that the expressions are equivalent, but this needs to be mathematically verified, which is the next step.

Step 3: Verify Algebraically

To further confirm our conjecture, we need to confirm it algebraically. This involves checking if \(y_{1}\) and \(y_{2}\) are equivalent expressions when simplified. Simplify both the expressions \(y_{1}\) and \(y_{2}\) fully and check if they have the same form. Use factoring, multiplying out or other algebraic techniques as needed.

Key Concepts

Equivalent Expressions

When solving algebraic equations, it is crucial to understand the concept of equivalent expressions. These are expressions that may look different but represent the same value for all values of the variable involved.

For example, the expressions \(2(x + 3)\) and \(2x + 6\) are equivalent because when you multiply out the first expression, you get the second expression. In our textbook exercise, we are dealing with more complex expressions: \(\frac{x^{4}+x^{2}-1}{x^{2}+1}\) and \(x^{2}-\frac{1}{x^{2}+1}\). The students' task is to use graphing to initially investigate whether these expressions are equivalent, and then use algebraic techniques to verify their equivalence.

Algebraic Verification

Algebraic verification is a definitive way to determine if two expressions are equivalent. After visually inspecting the graphs, the next step is to simplify the expressions algebraically. This involves combining like terms, factoring, expanding, and cancelling terms where applicable.

In the case of our example, you would take \(\frac{x^{4}+x^{2}-1}{x^{2}+1}\) and \(x^{2}-\frac{1}{x^{2}+1}\) and simplify them. If the simplified forms of both expressions are identical, we can conclude that \(y_{1}\) and \(y_{2}\) are in fact equivalent. Algebraic verification is precise and serves as a strong proof of equivalence.

Graphing Utility

A graphing utility is an important tool in visualizing algebraic equations and determining the relationship between them. With modern technology, students can input complex equations into graphing software or a graphing calculator to quickly see the relations and intersections of functions. For exercises involving graphing, like the one in question, it is recommended to use the zoom and trace features to closely observe whether the graphs of two functions lay perfectly on top of each other, suggesting equivalence.

However, while graphing utilities provide visual confirmation, they should be used in conjunction with algebraic verification to ensure accuracy, especially since visual checks can sometimes be misleading due to pixel resolution limits.

Function Simplification

Function simplification is a process of making a complex algebraic expression easier to work with by reducing it to its simplest form. This often involves eliminating parentheses, combining like terms, and reducing fractions. The goal is to express the function in the most straightforward way possible without altering its value.

In simplifying the functions \(y_{1}\) and \(y_{2}\) from the exercise, we would look for opportunities to factor, expand, and reduce the given expressions. Simplification can reveal the true nature of an expression and is essential for comparing two potential equivalent expressions, significantly aiding in algebraic verification.