Provide the complete sign analysis using a table to clearly indicate the behavior of the function when graphed between intervals. g(x) = 3²²14-4
Provide the complete sign analysis using a table to clearly indicate the behavior of the function when graphed between intervals. g(x) = 3²²14-4
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
![**Sign Analysis of the Function**
To understand the behavior of the function \( g(x) = \frac{x^2 - 4}{3x^2 + x - 4} \), we need to perform a sign analysis. This involves determining where the function is positive, negative, or undefined between intervals based on critical points.
### Function Analysis:
1. **Numerator: \( x^2 - 4 \)**
- Factor this as \( (x-2)(x+2) \).
- The numerator is zero at \( x = 2 \) and \( x = -2 \).
2. **Denominator: \( 3x^2 + x - 4 \)**
- To find the zeros of the denominator, solve \( 3x^2 + x - 4 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- Substituting \( a = 3, b = 1, c = -4 \):
\[
x = \frac{-1 \pm \sqrt{1 + 48}}{6} = \frac{-1 \pm \sqrt{49}}{6} = \frac{-1 \pm 7}{6}
\]
- This gives \( x = 1 \) and \( x = -\frac{4}{3} \).
### Critical Points:
- Zeros of the numerator: \( x = 2, x = -2 \)
- Zeros of the denominator: \( x = 1, x = -\frac{4}{3} \) (points of discontinuity)
### Intervals:
- \( (-\infty, -2) \)
- \( (-2, -\frac{4}{3}) \)
- \( (-\frac{4}{3}, 1) \)
- \( (1, 2) \)
- \( (2, \infty) \)
### Sign Analysis Table:
Evaluate the sign of the function within each interval:
- For each interval, choose a test point and determine the sign of \( g(x) \).
#### Conclusion:
Construct a table summarizing the results of the sign analysis:
| Interval | Test Point | Sign of \( g(x) \) |
|-----------------|](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc6a1e0b-5a8c-47c6-88a5-bfd01f5fa11f%2F6ed1c588-63a1-4717-be3c-f04cfb9454df%2Frzkzpv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Sign Analysis of the Function**
To understand the behavior of the function \( g(x) = \frac{x^2 - 4}{3x^2 + x - 4} \), we need to perform a sign analysis. This involves determining where the function is positive, negative, or undefined between intervals based on critical points.
### Function Analysis:
1. **Numerator: \( x^2 - 4 \)**
- Factor this as \( (x-2)(x+2) \).
- The numerator is zero at \( x = 2 \) and \( x = -2 \).
2. **Denominator: \( 3x^2 + x - 4 \)**
- To find the zeros of the denominator, solve \( 3x^2 + x - 4 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- Substituting \( a = 3, b = 1, c = -4 \):
\[
x = \frac{-1 \pm \sqrt{1 + 48}}{6} = \frac{-1 \pm \sqrt{49}}{6} = \frac{-1 \pm 7}{6}
\]
- This gives \( x = 1 \) and \( x = -\frac{4}{3} \).
### Critical Points:
- Zeros of the numerator: \( x = 2, x = -2 \)
- Zeros of the denominator: \( x = 1, x = -\frac{4}{3} \) (points of discontinuity)
### Intervals:
- \( (-\infty, -2) \)
- \( (-2, -\frac{4}{3}) \)
- \( (-\frac{4}{3}, 1) \)
- \( (1, 2) \)
- \( (2, \infty) \)
### Sign Analysis Table:
Evaluate the sign of the function within each interval:
- For each interval, choose a test point and determine the sign of \( g(x) \).
#### Conclusion:
Construct a table summarizing the results of the sign analysis:
| Interval | Test Point | Sign of \( g(x) \) |
|-----------------|
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