Problem 84 - Calculus of a Single Variable

Question

Comparing Functions In Exercises 83 and \(84,\) (a) use a graphing utility to graph \(f\) and \(g\) in the same viewing window, (b) verify algebraically that \(f\) and \(g\) represent the same function, and (c) zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.) $$ \begin{array}{l}{f(x)=-\frac{x^{3}-2 x^{2}+2}{2 x^{2}}} \\ {g(x)=-\frac{1}{2} x+1-\frac{1}{x^{2}}}\end{array} $$

Step-by-Step Solution

Answer

Both functions \(f(x)=-\frac{{x^{3}-2 x^{2}+2}}{{2 x^{2}}}\) and \(g(x)=-\frac{1}{2}x+1-\frac{1}{x^{2}}\) represent the same function and graph into the line \(y=-\frac{1}{2}x+1\) when sufficiently zoomed out on the graph.

Step 1: Graphing the functions

First, use a graphing utility like the online Desmos Graphing Calculator to graph both the functions \(f(x)\) and \(g(x)\). Enter the functions in separate lines and observe how they both appear on the graph.

Step 2: Algebraic verification

Next, to check algebraically if \(f(x)\) and \(g(x)\) represent the same function, simplify the equation of \(g(x)\) to its simplest form. By doing so you get \(g(x) = -\frac{1}{2}x + 1 - \frac{1}{x^2} = -\frac{x^3 - 2x^2 + 2}{2x^2}\) which is identical to the equation of \(f(x)\). Therefore, we can conclude that \(f(x)\) and \(g(x)\) represent the same function.

Step 3: Zoom out to line

If we zoom out far enough on the graph, both functions \(f(x)\) and \(g(x)\) will appear as a line. The slope of this line is -1/2 and intercepts the y-axis at 1, so the equation is \(y=-\frac{1}{2}x+1\). Do note that these observations may not be precise due to the limitations of visual assessments.

Key Concepts

Using a Graphing Utility

A graphing utility is an essential tool for visualizing mathematical functions. It helps you compare complex equations like the ones given in this exercise. By plotting both functions, \( f(x) \) and \( g(x) \), using an online graphing calculator such as Desmos, you can see if they align perfectly. When you enter the equations separately, observe how they overlap. This overlapping indicates that they may represent the same mathematical function. Graphing utilities provide a powerful visual method to support mathematical proofs and insights.

• Enhance understanding through visualization.
• Compare multiple functions quickly.
• Identify differences and similarities visually.

This tool enables immediate visual feedback and plays a crucial role in understanding complex relationships between functions.

Algebraic Verification

Algebraic verification involves analyzing equations to see if two functions are identical. In this exercise, you simplified \( g(x) \) to compare it directly with \( f(x) \). By simplifying \( g(x) = -\frac{1}{2}x + 1 - \frac{1}{x^2} \) to match \( f(x) = -\frac{x^3 - 2x^2 + 2}{2x^2} \), you discover that they simplify to the same expression.
This shows that they are indeed the same function. This step requires:

• Breaking down complex terms into simpler ones.
• Checking each term for equivalence.
• Confirming the overall expressions match.

Algebraic verification is critical in proving function identity without relying solely on visual methods.

Understanding Function Continuity

Function continuity is a concept that deals with how functions behave at every point in their domain. When zooming out on the graph of \( f(x) \) and \( g(x) \), they appear as a linear function \( y = -\frac{1}{2}x + 1 \).
However, the continuity nuances can be overlooked. If a function has points where it's not continuous, such as undefined points or jumps, these might not be visible when zoomed out. In this exercise, being aware that not all points are visible aids in understanding the hidden complexities within each function. Recognizing continuity:

• Helps predict function behavior.
• Aids in avoiding assumptions based on appearance.
• Ensures a thorough analysis.

Consider both visual cues and algebraic insights for a complete understanding of the functions.

Simplification of Functions

Simplifying functions is about rewriting them in a way that reveals more about their behavior. Simplification can make complex expressions more manageable and comparable. For instance, \( g(x) \) was simplified so it could be directly compared with \( f(x) \).
This resulted in understanding that both expressions were the same function. This simplification process involves:

• Reorganizing the terms in a function.
• Canceling out similar terms when possible.
• Making the overall structure simpler to interpret.

By simplifying functions, you can gain a clearer understanding of their properties and interactions, further aiding in insightful comparisons and analyses.