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Chapter 1: Problem 20

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. $$f(x)=\frac{x^{2}-1}{x^{2}+1}$$

Short Answer

Expert verified
Try graphing in the window \([-5, 5]\) for x and \([-2, 2]\) for y.

Step by step solution

01

Understand the Function

The given function is \( f(x) = \frac{x^2 - 1}{x^2 + 1} \). This function is a rational function where both the numerator and denominator are quadratic polynomials.
02

Analyze the Function's Behavior

To understand the behavior of the function, let's examine potential intercepts, asymptotes, and end behavior. - **Intercepts:** Set \( f(x) = 0 \) to find \( x \)-intercepts: \( x^2 - 1 = 0 \) gives solutions \( x = \pm 1 \). - **End Behavior:** As \( x \to \infty \) or \( x \to -\infty \), the function approaches 1 because the terms \( x^2 \) in the numerator and denominator dominate.
03

Determine the Vertical Asymptotes

There are no real vertical asymptotes since the denominator \( x^2 + 1 \) is always positive and never zero for real \( x \). Thus, the function is defined for all real \( x \).
04

Identify Symmetry

The function \( f(x) \) is symmetric with respect to the y-axis, as \( f(x) = f(-x) \). This tells us about the symmetric nature and can help narrow the viewing window.
05

Choose a Viewing Window

Considering the intercepts at \( x = \pm 1 \), symmetry, and the behavior as \( x \) approaches large values (function approaches 1), a potential viewing window is - **X-range:** \([-5, 5]\) - **Y-range:** \([-2, 2]\). This range captures all critical points and the horizontal asymptote.
06

Graph the Function

Using a graphing tool, input the function to see its behavior. Adjust the window if any important features, like intercepts or horizontal asymptotes, aren't visible or aren't clearly represented. Verify if the chosen window \([-5, 5]\) by \([-2, 2]\) adequately captures the behavior.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Rational functions are a fascinating part of algebra, involving ratios of two polynomials. In this exercise, our rational function is given by \( f(x) = \frac{x^2 - 1}{x^2 + 1} \).This equation features a numerator, \(x^2 - 1\), and a denominator, \(x^2 + 1\).

Rational functions can be complex, but a systematic approach helps! Begin by examining each element of the ratio:
  • The numerator \(x^2 - 1\) affects the zeroes of the function.
  • The denominator \(x^2 + 1\) influences where the function is undefined, if ever.
Given that polynomials in both numerator and denominator are quadratic, analyzing points where they zero out gives great insight. As you study rational functions, consider both polynomial degrees to understand limits and behaviors at infinity.
Function Intercepts
In graphing, intercepts show where a function crosses the axes. They are key to understanding a rational function's path. For \( f(x) = \frac{x^2 - 1}{x^2 + 1} \), determining intercepts starts with setting the numerator to zero.

Intercepts are calculated by:
  • Finding \(x\)-intercepts: Solve \(x^2 - 1 = 0\) to find \(x = \pm 1\). These are points where the graph crosses the \(x\)-axis.
  • Understanding \(y\)-intercepts: Set \(x = 0\) to calculate \(f(0)\), giving \(f(0) = -1\), meeting the graph on the y-axis.
As these intercepts guide your early sketches of the graph, they set the stage for identifying overall plot trends as you consider symmetry and asymptotes.
Asymptotic Behavior
Asymptotic behavior describes how function values behave when our input \(x\) becomes very large or small. Rational functions often have asymptotes, horizontal, vertical, or slanted, helping us see what values are unattainable or approached at extremes.

In our function, \(f(x) = \frac{x^2 - 1}{x^2 + 1} \), observing asymptotic behavior involves:
  • Horizontal Asymptotes: As \(x\rightarrow \pm \infty\), the terms \(x^2\) rule the expression, leading \(f(x)\rightarrow 1\). This implies a horizontal asymptote at \(y = 1\).
  • Vertical Asymptotes: Absent here since \(x^2 + 1\) never becomes zero for real \(x\), keeping our function defined across real numbers.
This asymptotic prediction guides us in understanding that our function stays steadily near \(y=1\) for large \(x\) values, without any dramatic deviations.
Graphing Window Selection
Choosing an effective viewing window is crucial. It influences how well you can observe a function's details and behavior when graphing, especially for rational functions like \(f(x) = \frac{x^2 - 1}{x^2 + 1} \).

Effective window selection involves:
  • Identifying critical points: This includes intercepts and symmetry, such as solutions \(x = \pm 1\) and \(f(0) = -1\).
  • Factor in asymptotic behavior: Note the horizontal asymptote at \(y=1\).
  • Selecting sensible ranges: X-range of \([-5, 5]\) and Y-range of \([-2, 2]\) effectively encapsulates the significant elements and overall function behavior.
A well-chosen window reveals the graph's intercepts, ensures visibility of trends around \(y = 1\), and provides a coherent view of the graph's full scope in the selected range.

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Most popular questions from this chapter

Tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. \(y=1+\frac{1}{x^{2}}, \quad\) compressed vertically by a factor of 2.

Say whether the function is even, odd, or neither. Give reasons for your answer. $$h(t)=2|t|+1$$

You will explore graphically the general sine function $$f(x)=A \sin \left(\frac{2 \pi}{B}(x-C)\right)+D$$ as you change the values of the constants \(A, B, C,\) and \(D .\) Use a CAS or computer grapher to perform the steps in the exercises. Set the constants \(A=3, B=6, D=0\). a. Plot \(f(x)\) for the values \(C=0,1,\) and 2 over the interval \(-4 \pi \leq x \leq 4 \pi .\) Describe what happens to the graph of the general sine function as \(C\) increases through positive values. b. What happens to the graph for negative values of \(C ?\) c. What smallest positive value should be assigned to \(C\) so the graph exhibits no horizontal shift? Confirm your answer with a plot.

Graph the functions. $$y=|1-x|-1.$$

Tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. \(y=1-x^{3},\) compressed horizontally by a factor of 3.
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