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 1.** Graph the function \( f(x) = \frac{x^2 + 6x + 5}{x - 1} \)
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The function \( f(x) = \frac{x^2 + 6x + 5}{x - 1} \) is a rational function. To graph this function, follow these steps:
1. **Factor the numerator:** Identify and factor any common terms in the numerator.
\[ x^2 + 6x + 5 = (x + 1)(x + 5) \]
2. **Rewrite the function:** Substitute the factored form into the original function.
\[ f(x) = \frac{(x + 1)(x + 5)}{x - 1} \]
3. **Identify vertical asymptotes:** Set the denominator equal to zero to find the x-values at which the function is undefined.
\[ x - 1 = 0 \implies x = 1 \]
Therefore, there is a vertical asymptote at \( x = 1 \).
4. **Find oblique asymptotes or holes:**
- To determine if there are any holes, look for common factors in the numerator and denominator. Since \( (x - 1) \) is not a factor of the numerator, there are no holes in the graph.
- To find the oblique asymptote, perform polynomial long division of \( x^2 + 6x + 5 \) by \( x - 1 \).
\[ \begin{array}{r|ll}
x + 7 & x - 1 \big( x^2 + 6x + 5 \\
& - (x^2 - x) \\
& \phantom{xxxxx}6x + x \\
& \phantom{xxxxx} - (6x - 6) \\
& \phantom{xxxxxxxx}11 \\
\end{array} \]
The quotient is \( x + 7 \), and the remainder is 11. Thus, there is an oblique (slant) asymptote given by the equation:
\[ y = x + 7 \]
5. **Find the x-intercepts:** Set \( f(x) = 0 \).
\[ \frac{(x + 1)(x + 5)}{x -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4526b347-f4c2-4959-b519-c79fc40f5b52%2F87756432-6f1f-4e73-8ae3-425984ddd644%2F5gcl5n8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1.** Graph the function \( f(x) = \frac{x^2 + 6x + 5}{x - 1} \)
---
The function \( f(x) = \frac{x^2 + 6x + 5}{x - 1} \) is a rational function. To graph this function, follow these steps:
1. **Factor the numerator:** Identify and factor any common terms in the numerator.
\[ x^2 + 6x + 5 = (x + 1)(x + 5) \]
2. **Rewrite the function:** Substitute the factored form into the original function.
\[ f(x) = \frac{(x + 1)(x + 5)}{x - 1} \]
3. **Identify vertical asymptotes:** Set the denominator equal to zero to find the x-values at which the function is undefined.
\[ x - 1 = 0 \implies x = 1 \]
Therefore, there is a vertical asymptote at \( x = 1 \).
4. **Find oblique asymptotes or holes:**
- To determine if there are any holes, look for common factors in the numerator and denominator. Since \( (x - 1) \) is not a factor of the numerator, there are no holes in the graph.
- To find the oblique asymptote, perform polynomial long division of \( x^2 + 6x + 5 \) by \( x - 1 \).
\[ \begin{array}{r|ll}
x + 7 & x - 1 \big( x^2 + 6x + 5 \\
& - (x^2 - x) \\
& \phantom{xxxxx}6x + x \\
& \phantom{xxxxx} - (6x - 6) \\
& \phantom{xxxxxxxx}11 \\
\end{array} \]
The quotient is \( x + 7 \), and the remainder is 11. Thus, there is an oblique (slant) asymptote given by the equation:
\[ y = x + 7 \]
5. **Find the x-intercepts:** Set \( f(x) = 0 \).
\[ \frac{(x + 1)(x + 5)}{x -
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