Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find any asymptotes of the graph of the rational function. Verify your answers by using a graphing utility to graph the function. $$f(x)=\frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

The vertical asymptote of the function \(f(x)=\frac{1}{x^{2}}\) is at \(x = 0\) and the horizontal asymptote is at \(y = 0\).

Step 1: Find the Vertical Asymptotes

The denominator of the function is \(x^{2}\). Set \(x^{2} = 0\) and solve for \(x\). This will give the vertical asymptotes.

Step 2: Find the Horizontal Asymptotes

The degree of the numerator is 0 (because 1 is equal to \(x^{0}\)) while the degree of the denominator is 2. This means that the degree of the numerator is less than the degree of the denominator and as such \(y = 0\) is a horizontal asymptote.

Step 3: Verify with a Graphing Utility

Using a graphing utility, input the function \(f(x)=\frac{1}{x^{2}}\). The graph will reveal vertical asymptotes where \(x = 0\) and a horizontal asymptote at \(y = 0\).

Key Concepts

Vertical Asymptotes

In the realm of rational functions, vertical asymptotes occur where the denominator equals zero. They represent values that the function cannot cross or meet. These are the steep lines on the graph where the function appears to shoot up or down infinitely. To find vertical asymptotes, set the denominator of the function equal to zero and solve for the variable.

• Apply this to the function \(f(x) = \frac{1}{x^2}\), and you'll set \(x^2 = 0\).
• This results in \(x = 0\).

The function \(f(x)\) has a vertical asymptote at \(x=0\). This means the function does not touch or cross the line \(x=0\). Instead, it diverges as it approaches this point from either direction. Vertical asymptotes are crucial for understanding the domain of a function, especially in cases where calculus is applied to investigate limits and continuity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \(x\) approaches infinity or negative infinity. They help visualize the trend of the function as it stretches towards extreme ends on the x-axis. To find them, compare the degrees of the numerator and denominator in a rational function.

• If the degree of the numerator is less than the degree of the denominator, like in \(f(x) = \frac{1}{x^2}\), the horizontal asymptote is \(y = 0\).
• This happens because the value of \(y\) gets closer to zero as \(x\) becomes very large or very small.

In simpler terms, as you move away from the origin in either direction, the function \(f(x)\) gets closer and closer to the x-axis but never actually touches it. Horizontal asymptotes tell us about the end behavior of the graph, which is especially useful in calculus when determining limits.

Graphing Utility

A graphing utility, like a graphing calculator or computer software, is a valuable tool in visualizing mathematical functions. They allow you to see the general shape and specific behaviors of a function. Verifying answers with a graphing utility provides an additional layer of understanding.

• By entering the function \(f(x) = \frac{1}{x^2}\) into the graphing utility, you can visually confirm the presence of asymptotes detected in calculations.
• The graph should clearly show a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).
• This visual approach helps verify and solidify your analytical findings.

Graphing utilities are especially helpful for students to intuitively grasp the behavior of complex functions. These tools bring mathematical concepts to life, making them easier to comprehend and apply in various contexts.