Finding a Viewing Window


In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s 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.


f(x) = (x² − 1)/(x² + 1)

Verified step by step guidance

1. Step 1: Analyze the function f(x) = (x² − 1)/(x² + 1) to understand its behavior. Notice that both the numerator and the denominator are quadratic polynomials. The function is defined for all real numbers since the denominator x² + 1 is never zero.
2. Step 2: Determine the horizontal asymptote by considering the limits as x approaches infinity. Since the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients, which is 1. Therefore, y = 1 is a horizontal asymptote.
3. Step 3: Find the x-intercepts by setting the numerator equal to zero: x² - 1 = 0. Solving this gives x = ±1. These are the points where the graph crosses the x-axis.
4. Step 4: Evaluate the behavior near the y-axis by finding the y-intercept. Set x = 0 in the function: f(0) = (0² - 1)/(0² + 1) = -1. This gives the y-intercept at (0, -1).
5. Step 5: Choose a viewing window that includes the x-intercepts, the y-intercept, and the horizontal asymptote. A suitable window might be x: [-3, 3] and y: [-2, 2] to capture the overall behavior of the function, including the intercepts and the asymptotic behavior.

Key Concepts

Function Behavior Analysis

Understanding the behavior of a function involves analyzing its key features such as intercepts, asymptotes, and end behavior. For the function f(x) = (x² − 1)/(x² + 1), examining these features helps determine how the function behaves across different values of x, which is crucial for selecting an appropriate viewing window that captures the function's overall behavior.

Graphing Software Viewing Window

A viewing window in graphing software defines the range of x and y values displayed on the graph. Choosing the right window is essential to accurately represent the function's behavior, ensuring that important features like peaks, valleys, and asymptotes are visible. This involves setting appropriate limits for x and y axes based on the function's characteristics.

Rational Functions

Rational functions are ratios of polynomials, and their graphs can exhibit unique features such as vertical and horizontal asymptotes. For f(x) = (x² − 1)/(x² + 1), understanding how the numerator and denominator affect the graph is key. The function's behavior near asymptotes and its end behavior as x approaches infinity or negative infinity are critical for determining the viewing window.