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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Question
please make sure give the intercepts of the equation.
![**Finding Asymptotes of Rational Functions**
In this exercise, we analyze the function \( g(x) = \frac{3x - 4}{x + 2} \) to find its asymptotes and express them as linear equations.
1. **Vertical Asymptote**:
- Vertical asymptotes occur where the denominator is zero (and the numerator is non-zero).
- Set the denominator equal to zero:
\[
x + 2 = 0
\]
- Solving for \( x \), we find \( x = -2 \).
- Thus, there is a vertical asymptote at \( x = -2 \).
2. **Horizontal Asymptote**:
- Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator.
- The degrees of both the numerator \( (3x - 4) \) and the denominator \( (x + 2) \) are 1.
- When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.
- The leading coefficient of the numerator is 3, and for the denominator, it is 1.
- Hence, the horizontal asymptote is:
\[
y = \frac{3}{1} = 3
\]
Therefore, the function \( g(x) = \frac{3x - 4}{x + 2} \) has a vertical asymptote at \( x = -2 \) and a horizontal asymptote at \( y = 3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49520758-34b8-4feb-b56e-1bcd3eb1e5da%2F8dd8ff3d-551d-4ba0-9c17-1855afca0e4b%2Fklgsd05_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding Asymptotes of Rational Functions**
In this exercise, we analyze the function \( g(x) = \frac{3x - 4}{x + 2} \) to find its asymptotes and express them as linear equations.
1. **Vertical Asymptote**:
- Vertical asymptotes occur where the denominator is zero (and the numerator is non-zero).
- Set the denominator equal to zero:
\[
x + 2 = 0
\]
- Solving for \( x \), we find \( x = -2 \).
- Thus, there is a vertical asymptote at \( x = -2 \).
2. **Horizontal Asymptote**:
- Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator.
- The degrees of both the numerator \( (3x - 4) \) and the denominator \( (x + 2) \) are 1.
- When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.
- The leading coefficient of the numerator is 3, and for the denominator, it is 1.
- Hence, the horizontal asymptote is:
\[
y = \frac{3}{1} = 3
\]
Therefore, the function \( g(x) = \frac{3x - 4}{x + 2} \) has a vertical asymptote at \( x = -2 \) and a horizontal asymptote at \( y = 3 \).
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