Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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find the intervals on which f is concave up or concave down
![### Exploring Functions: Rational Functions
Let's consider the function \( f(x) \) defined by
\[ f(x) = \frac{1}{1 - x^2}. \]
This function represents a rational function, where the numerator is a constant (1) and the denominator is a quadratic polynomial \( 1 - x^2 \).
#### Key Concepts:
1. **Domain**: Identify the values of \( x \) for which the function is defined. Since division by zero is undefined, set the denominator \( 1 - x^2 = 0 \) and solve for \( x \). This gives \( x = \pm 1 \). Therefore, the function is undefined at \( x = 1 \) and \( x = -1 \).
2. **Vertical Asymptotes**: The values of \( x \) that make the denominator zero and are not canceled out by the numerator. In this case, \( x = 1 \) and \( x = -1 \) are vertical asymptotes.
3. **Behavior at Asymptotes**: As \( x \) approaches 1 or -1, the function values tend to infinity or negative infinity. This can be observed by taking limits:
\[ \lim_{x \to 1^{-}} f(x) = -\infty, \quad \lim_{x \to 1^{+}} f(x) = \infty \]
\[ \lim_{x \to -1^{-}} f(x) = \infty, \quad \lim_{x \to -1^{+}} f(x) = -\infty \]
4. **Even Function**: The function \( f(x) \) is symmetric about the y-axis because the expression \( f(x) = f(-x) \).
#### Graph of \( f(x) \):
The graph of \( f(x) = \frac{1}{1-x^2} \) would show two vertical asymptotes at \( x = 1 \) and \( x = -1 \). Between these asymptotes, the graph has two separate parts, each moving towards infinity as \( x \) approaches 1 or -1.
For visual representation:
- For \( x \) in the interval \( (-\infty, -1) \), \( f(x) \) decreases from 0 to negative infinity as \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc931b694-aaff-47c2-895e-73bc1d52993a%2F446c0743-15e6-4f87-ab85-bf64c323f26b%2Faabvyq4_processed.png&w=3840&q=75)
Transcribed Image Text:### Exploring Functions: Rational Functions
Let's consider the function \( f(x) \) defined by
\[ f(x) = \frac{1}{1 - x^2}. \]
This function represents a rational function, where the numerator is a constant (1) and the denominator is a quadratic polynomial \( 1 - x^2 \).
#### Key Concepts:
1. **Domain**: Identify the values of \( x \) for which the function is defined. Since division by zero is undefined, set the denominator \( 1 - x^2 = 0 \) and solve for \( x \). This gives \( x = \pm 1 \). Therefore, the function is undefined at \( x = 1 \) and \( x = -1 \).
2. **Vertical Asymptotes**: The values of \( x \) that make the denominator zero and are not canceled out by the numerator. In this case, \( x = 1 \) and \( x = -1 \) are vertical asymptotes.
3. **Behavior at Asymptotes**: As \( x \) approaches 1 or -1, the function values tend to infinity or negative infinity. This can be observed by taking limits:
\[ \lim_{x \to 1^{-}} f(x) = -\infty, \quad \lim_{x \to 1^{+}} f(x) = \infty \]
\[ \lim_{x \to -1^{-}} f(x) = \infty, \quad \lim_{x \to -1^{+}} f(x) = -\infty \]
4. **Even Function**: The function \( f(x) \) is symmetric about the y-axis because the expression \( f(x) = f(-x) \).
#### Graph of \( f(x) \):
The graph of \( f(x) = \frac{1}{1-x^2} \) would show two vertical asymptotes at \( x = 1 \) and \( x = -1 \). Between these asymptotes, the graph has two separate parts, each moving towards infinity as \( x \) approaches 1 or -1.
For visual representation:
- For \( x \) in the interval \( (-\infty, -1) \), \( f(x) \) decreases from 0 to negative infinity as \(
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