x² - 1 Consider the following function: f(x)= Find the equation for Зx2 — 3х — 6 each horizontal asymptote and each vertical asymptote. Justify your answer. Find the x- and y-intercepts. Find the domain.
x² - 1 Consider the following function: f(x)= Find the equation for Зx2 — 3х — 6 each horizontal asymptote and each vertical asymptote. Justify your answer. Find the x- and y-intercepts. Find the domain.
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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![### Problem Statement
Consider the following function:
\[ f(x) = \frac{x^2 - 1}{3x^2 - 3x - 6} \]
1. **Find the equation for each horizontal asymptote and each vertical asymptote. Justify your answer.**
2. **Find the \( x \)- and \( y \)-intercepts.**
3. **Find the domain.**
### Instructions for Analysis
#### 1. Asymptote Analysis
- **Horizontal Asymptote:** To determine the horizontal asymptote, compare the degrees of the numerator and the denominator of the function \( f(x) \).
- **Vertical Asymptote:** To find the vertical asymptote, set the denominator equal to zero and solve for \( x \).
#### 2. Intercept Analysis
- **\( x \)-intercepts:** Set the numerator equal to zero and solve for \( x \).
- **\( y \)-intercepts:** Evaluate \( f(0) \).
#### 3. Domain Analysis
- Identify the values of \( x \) for which the function \( f(x) \) is undefined (i.e., where the denominator is zero).
### Detailed Steps and Solutions
**Horizontal Asymptote:**
- The degrees of the numerator and the denominator are both 2. The horizontal asymptote is found by taking the leading coefficients of \( x^2 \) from both the numerator and the denominator. Thus, the horizontal asymptote is:
\[ y = \frac{1}{3}. \]
**Vertical Asymptote:**
- Set the denominator equal to zero and solve for \( x \):
\[ 3x^2 - 3x - 6 = 0. \]
- Factor the quadratic equation:
\[ 3(x^2 - x - 2) = 0 \implies 3(x - 2)(x + 1) = 0. \]
- The solutions are:
\[ x = 2 \quad \text{and} \quad x = -1. \]
Thus, the vertical asymptotes are:
\[ x = 2 \quad \text{and} \quad x = -1. \]
**\( x \)-intercepts:**
- Set the numerator equal to zero and solve for \( x \):
\[ x^2 - 1 = 0 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa55013e1-91aa-48be-9c4b-bf8a0e47de22%2F3259dbf7-ad60-4ec2-93eb-a75fd4999062%2Fj6jzp4f_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Consider the following function:
\[ f(x) = \frac{x^2 - 1}{3x^2 - 3x - 6} \]
1. **Find the equation for each horizontal asymptote and each vertical asymptote. Justify your answer.**
2. **Find the \( x \)- and \( y \)-intercepts.**
3. **Find the domain.**
### Instructions for Analysis
#### 1. Asymptote Analysis
- **Horizontal Asymptote:** To determine the horizontal asymptote, compare the degrees of the numerator and the denominator of the function \( f(x) \).
- **Vertical Asymptote:** To find the vertical asymptote, set the denominator equal to zero and solve for \( x \).
#### 2. Intercept Analysis
- **\( x \)-intercepts:** Set the numerator equal to zero and solve for \( x \).
- **\( y \)-intercepts:** Evaluate \( f(0) \).
#### 3. Domain Analysis
- Identify the values of \( x \) for which the function \( f(x) \) is undefined (i.e., where the denominator is zero).
### Detailed Steps and Solutions
**Horizontal Asymptote:**
- The degrees of the numerator and the denominator are both 2. The horizontal asymptote is found by taking the leading coefficients of \( x^2 \) from both the numerator and the denominator. Thus, the horizontal asymptote is:
\[ y = \frac{1}{3}. \]
**Vertical Asymptote:**
- Set the denominator equal to zero and solve for \( x \):
\[ 3x^2 - 3x - 6 = 0. \]
- Factor the quadratic equation:
\[ 3(x^2 - x - 2) = 0 \implies 3(x - 2)(x + 1) = 0. \]
- The solutions are:
\[ x = 2 \quad \text{and} \quad x = -1. \]
Thus, the vertical asymptotes are:
\[ x = 2 \quad \text{and} \quad x = -1. \]
**\( x \)-intercepts:**
- Set the numerator equal to zero and solve for \( x \):
\[ x^2 - 1 = 0 \
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