f(x) = x²+6x+5 2-1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Find the inflection points
![**Mathematical Function**
The given function is:
\[ f(x) = \frac{x^2 + 6x + 5}{x - 1} \]
This function is a rational function where the numerator is a quadratic polynomial (\(x^2 + 6x + 5\)) and the denominator is a linear polynomial (\(x - 1\)).
**Key Points to Note:**
- **Zeros of the Function:** The zeros of the function can be found by setting the numerator equal to zero and solving for \(x\).
\[ x^2 + 6x + 5 = 0 \]
Factoring the quadratic polynomial:
\[ (x + 1)(x + 5) = 0 \]
Hence, the zeros are:
\[ x = -1 \]
\[ x = -5 \]
- **Vertical Asymptote:** A vertical asymptote occurs where the denominator of the rational function is zero (and the numerator is not zero at that point). For this function, the vertical asymptote is at:
\[ x = 1 \]
- **Horizontal Asymptote:** Since the degree of the numerator (2) is higher than the degree of the denominator (1), there is no horizontal asymptote. Instead, the end behavior of the function will resemble that of the polynomial that remains when the rational function is divided.
Additionally, if the polynomial can be simplified further, it may reveal more details about the behavior of the function.
Feel free to explore more about the graphical representation and the behavior of the function around these critical points. Understanding rational functions is fundamental in calculus and advanced algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4526b347-f4c2-4959-b519-c79fc40f5b52%2Fdd2c9c43-c06a-4a80-909f-60b8014dc660%2F1oruj3_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Function**
The given function is:
\[ f(x) = \frac{x^2 + 6x + 5}{x - 1} \]
This function is a rational function where the numerator is a quadratic polynomial (\(x^2 + 6x + 5\)) and the denominator is a linear polynomial (\(x - 1\)).
**Key Points to Note:**
- **Zeros of the Function:** The zeros of the function can be found by setting the numerator equal to zero and solving for \(x\).
\[ x^2 + 6x + 5 = 0 \]
Factoring the quadratic polynomial:
\[ (x + 1)(x + 5) = 0 \]
Hence, the zeros are:
\[ x = -1 \]
\[ x = -5 \]
- **Vertical Asymptote:** A vertical asymptote occurs where the denominator of the rational function is zero (and the numerator is not zero at that point). For this function, the vertical asymptote is at:
\[ x = 1 \]
- **Horizontal Asymptote:** Since the degree of the numerator (2) is higher than the degree of the denominator (1), there is no horizontal asymptote. Instead, the end behavior of the function will resemble that of the polynomial that remains when the rational function is divided.
Additionally, if the polynomial can be simplified further, it may reveal more details about the behavior of the function.
Feel free to explore more about the graphical representation and the behavior of the function around these critical points. Understanding rational functions is fundamental in calculus and advanced algebra.
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