Problem 20 - Thomas Calculus

Question

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. \begin{equation} f(x)=\frac{x^{2}-1}{x^{2}+1} \end{equation}

Step-by-Step Solution

Answer

The viewing window \([-10, 10] \) for \( x \) and \([-2, 2] \) for \( y \) shows the function's important features.

Step 1: Analyze the Function

The given function is \( f(x) = \frac{x^{2} - 1}{x^{2} + 1} \). We need to understand its behavior first. Notice that both the numerator \( x^2 - 1 \) and the denominator \( x^2 + 1 \) are quadratic expressions. The numerator factors as \( (x - 1)(x + 1) \) and does not cause division by zero, since the denominator \( x^2 + 1 \) is always positive. As \( x \to \pm\infty \), \( f(x) \to 1 \).

Step 2: Identify Intercepts and Asymptotes

Determine the x-intercepts by setting the numerator equal to zero: \( x^2 - 1 = 0 \) which gives \( x = \pm 1 \). On the other hand, the y-intercept is calculated by evaluating \( f(0) = \frac{-1}{1} = -1 \). There are no vertical asymptotes since the denominator never equals zero, and as \( x \to \pm\infty \), \( f(x) \to 1 \), indicating a horizontal asymptote at \( y = 1 \).

Step 3: Choose an Initial Window

A good initial viewing window for the graph might be \([-10, 10]\) for \( x \) and \([-2, 2]\) for \( y \). This range should capture the intercepts and the horizontal asymptote. However, it might need adjusting after checking the graph.

Step 4: Plot the Function

Use graphing software to plot \( f(x) \) within the window \([-10, 10] \) for \( x \) and \([-2, 2] \) for \( y \). Ensure that all key features are visible: the x-intercepts at \( x = \pm 1 \), the y-intercept at \( y = -1 \), and the horizontal asymptote at \( y = 1 \).

Step 5: Adjust the Window if Necessary

If the initial graph doesn't clearly show these features, adjust the window. Consider expanding \( x \) to better view the approach to the horizontal asymptote or modify the \( y \) bounds if the curve near the x-intercepts isn't clear enough.

Key Concepts

Function Behavior Analysis

When looking at a mathematical function, it's crucial to analyze how it behaves as its input value, or "x," changes. The function from our example, \( f(x) = \frac{x^2 - 1}{x^2 + 1} \), is a rational function. When analyzing its behavior, check both the numerator and denominator. Here, both are quadratic expressions, and the denominator, \( x^2 + 1 \), is always positive. This means that the function is defined for all x values, with no undefined points that create division by zero.

A significant part of function analysis is understanding intercepts and asymptotic behavior:

• **X-Intercepts:** Solve \( x^2 - 1 = 0 \) to get \( x = \pm 1 \).
• **Y-Intercept:** Substitute \( x = 0 \) in \( f(x) \) giving \( f(0) = -1 \).
• **Horizontal Asymptote:** Examine what happens as \( x \to \pm \infty \). The expression approaches 1, indicating a horizontal asymptote at \( y = 1 \).

This understanding is necessary before you begin graphing to predict what the graph should look like and to choose appropriate viewing windows.

Viewing Window Adjustment

Using graphing software requires choosing a good initial viewing window that effectively shows the important features of the function graph. Adjust the window settings by zooming in or out to ensure all critical behaviors are visible.

For the function \( f(x) = \frac{x^2 - 1}{x^2 + 1} \), an initial proposal could be:

• For the **x-axis**, the range \([-10, 10]\) is broad enough to cover the necessary x-intercepts at \( x = \pm 1 \).
• For the **y-axis**, the range \([-2, 2]\) captures the y-intercept at \( y = -1 \) and shows the behavior approaching the horizontal asymptote at \( y = 1 \).

If the first attempt doesn't show all these features clearly, don't hesitate to adjust! Try expanding the x-range past 10 or tweaking y-bounds to more effectively visualize key parts of the function. This iterative process ensures a complete view of the function's behavior.

Horizontal Asymptote Identification

A horizontal asymptote is essentially a line that a function approaches as its input grows indefinitely large (positive or negative). For rational functions like \( f(x) = \frac{x^2 - 1}{x^2 + 1} \), determine asymptotes by comparing the degrees of the numerator and denominator.

In this example, both the numerator \( x^2 - 1 \) and the denominator \( x^2 + 1 \) are quadratic, which means they share the same degree. According to asymptote rules, if the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients (1/1 in this case), resulting in \( y = 1 \).

This horizontal line represents the value to which \( f(x) \) will get very close as \( x \to \infty \) or \( x \to -\infty \). While the graph may never touch or cross this line in such scenarios, identifying asymptotes is critical for understanding the end behavior of the function, which impacts the choice of graph viewing windows and the interpretation of the graph itself.