Problem 76 - Calculus of a Single Variable

Question

Analyzing a Graph Using Technology In Exercises \(75-82,\) use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{1}{x^{2}-x-2} $$

Step-by-Step Solution

Answer

The vertical asymptotes of the function are \(x=-1\) and \(x=2\), and the horizontal asymptote is \(y=0\). The function has no extrema.

Step 1: Calculate the Vertical Asymptotes

The vertical asymptotes can be found by determining the values of x for which the function is undefined. This happens when the denominator of the function is zero. Thus, solve the equation \(x^{2}-x-2=0\). This yields \(x=-1\) and \(x=2\). These are the vertical asymptotes.

Step 2: Calculate the Horizontal Asymptote

The horizontal asymptote can be found by taking the limit of the function as x approaches positive and negative infinity. In this case, observe that as \(x\) grows large or becomes very negative, the \(x^{2}\) term in the denominator dominates the other terms, thus the function tends to zero. Hence, y = 0 is the horizontal asymptote.

Step 3: Determine the Extrema

The function does not have maximum or minimum points within the real number domain because it is a rational function and it doesn't have turning points of the type encountered in polynomial functions.

Key Concepts

Vertical Asymptotes

A vertical asymptote is a line that a graph approaches but never crosses. It occurs where a function's value becomes unbounded, typically because of division by zero in the denominator. For the function \(f(x) = \frac{1}{x^{2} - x -2}\), we need to find when the denominator equals zero. This informs us where the function is undefined and helps locate vertical asymptotes.

Follow these steps:

• Set the denominator of the function, \(x^2 - x - 2 = 0\), and solve for \(x\).
• Factor or use the quadratic formula to find \(x = -1\) and \(x = 2\).

These values, \(x = -1\) and \(x = 2\), are the vertical asymptotes. They are crucial for understanding the boundaries and behavior of the function as the graph approaches these lines but never intersects them. Each side of the vertical asymptote behaves differently, showing how graphs can stretch towards infinity at these lines.

Horizontal Asymptotes

Horizontal asymptotes provide insight into the behavior of a function as \(x\) approaches positive or negative infinity. They show the value that a function approaches as it "flattens out." In the function \(f(x) = \frac{1}{x^{2} - x -2}\), we discover the horizontal asymptote by examining what happens as \(x\) becomes very large or very negative.

Here’s how you analyze it:

• Consider the highest degree of \(x\) in the denominator, which is \(x^2\).
• At large values of \(x\), \(x^2\) dominates, making the function approach zero.

Thus, the function approaches \(y = 0\) on both ends of the x-axis. This horizontal asymptote means the graph gets arbitrarily close to the x-axis but never quite reaches it as \(x\) becomes increasingly large or small. It's a good indication that as you zoom out, the graph flattens towards this line.

Extrema

In mathematics, extrema refer to the maximum and minimum values a function can attain. These are critical for understanding the high or low points in the graph of a function. For rational functions like \(f(x) = \frac{1}{x^{2} - x -2}\), extrema may not always exist like in polynomial functions.

Here’s why this function doesn't have extrema:

• Rational functions can have turning points only if they contain polynomials of certain degrees.
• This specific function lacks such turning points, meaning there are no peaks or valleys where the derivative equals zero in its domain.

Hence, for \(f(x)\), there are no points of maximum or minimum within the real number scope due to its rational nature. Recognizing when extrema are present or not is key to graph analysis, helping to understand the overall shape and behavior of the function.