Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to (a) graph the function and (b) find any asymptotes numerically by creating a table of values for the function. $$f(x)=-\frac{6}{2-e^{0.2 x}}$$

Step-by-Step Solution

Answer

Use a graphing utility to graph the function and create a table of values to help determine any asymptotes numerically.

Step 1: Graphing the function

To graph \(f(x)= -\frac{6}{2-e^{0.2 x}}\), first plot basic points through the graphing utility. The graphing utility will be able to show a visual representation of the function.

Step 2: Creating a table

Secondly, create a table of values using the graphing utility. This is usually done by inputting ranges for x-values, and letting the utility compute the corresponding y-values or \(f(x)\) for the function.

Step 3: Finding asymptotes numerically

By closely examining the table, you can determine if there are any values of x for which the function approaches infinity or negative infinity, thereby identifying any vertical asymptotes. Horizontal asymptotes can be found by looking at the behavior of the function at extreme positive and negative x-values. If the function seems to approach a certain value, then there is a horizontal asymptote.

Key Concepts

Graphing Utility

Using a graphing utility can be a powerful tool when visualizing complex functions. These tools allow you to input any mathematical function, like \( f(x) = -\frac{6}{2-e^{0.2x}} \), and see its graph on a coordinate plane. This visual representation helps in understanding the shape and behavior of the graph.

Graphing utilities are user-friendly and typically require you to input the function as it is. Then, with just a click, the tool will plot the function, showing crucial features such as peaks, valleys, and asymptotic behavior.

• Ease of use: Simply enter the equation and let the utility do the work.
• Visual aid: It provides a clear and immediate visual of the function's behavior.
• Dynamic exploration: Many utilities allow manipulation of variables to see how changes affect the graph.

Using a graphing utility makes it easier to see where a function might be undefined or approaching certain values as asymptotes.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. They are essential in understanding the behavior of rational functions like \( f(x) = -\frac{6}{2-e^{0.2x}} \). Asymptotes can be vertical or horizontal, and sometimes even slant (though we focus on the former two here).

In our function, vertical asymptotes occur where the function is undefined, which can typically be found where the denominator equals zero. So, solving \( 2 - e^{0.2x} = 0 \) will give insight into possible vertical asymptotes.
Horizontal asymptotes are determined by observing the function as \( x \) approaches infinity or negative infinity. For the provided function, if we see that \( f(x) \) is approaching a certain value, then that value indicates the location of a horizontal asymptote.

• Vertical asymptotes occur at values making the denominator zero.
• Horizontal asymptotes are found by analyzing the limits of the function as \( x \to \infty \) or \( x \to -\infty \).

Understanding asymptotes helps in graphing functions and predicting end behavior.

Table of Values

Creating a table of values is an effective method to complement the visual representation of a function graph. This numerical approach aids in recognizing patterns and behaviors of \( f(x) \), such as where it might head towards infinity or any possible asymptotic behavior.

To create a table of values for the function \( f(x) = -\frac{6}{2-e^{0.2x}} \), choose a range for \( x \), input these into the graphing utility, and record the resulting \( f(x) \) values. The table will list pairs of \( (x, f(x)) \), showcasing how \( f(x) \) changes with respect to \( x \).

• Aids in identifying vertical asymptotes by revealing rapid increases or decreases in \( f(x) \).
• Helps to derive horizontal asymptotes by observing \( f(x) \) trends as \( x \) increases or decreases significantly.
• Provides precise numerical data to support the graphing utility's visual output.

Using a table of values bridges the gap between numeric results and graphical representation, making it easier to comprehend the complete behavior of a function.