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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 1.4.20
Capitolo 1, Problema 1.4.20

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)

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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.
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.
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.
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).
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.

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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.
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Graphs of Exponential Functions

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.
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Example 4: Norman Window

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.
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Intro to Rational Functions
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Graphing


In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.


y = (−2x)²/³

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Radians and Degrees


On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?

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A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.

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Finding Formulas for Functions


A point P in the first quadrant lies on the graph of the function f(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.

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Piecewise-Defined Functions


In Exercises 35 and 36, find the (a) domain and (b) range.



𝔂 = { √ -x, -4 ≤ x ≤ 0

{ √ x, 0 < x ≤ 4

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Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?


a. ƒg

b. ƒ³

c. ƒ(sin x)

d. g(sec x)

e. |g|

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