Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Find all Horizontal Asymptotes
![**Problem 53: Rational Function Analysis**
Given the rational function \( f(x) = \frac{(2x - 5)^2}{x^2 - x} \), we are to analyze its behavior and properties.
**Function Description:**
- **Numerator:** \((2x - 5)^2\)
- **Denominator:** \(x^2 - x\)
This function can be broken down into the product and division of polynomial functions. Let's delve deeper into its characteristics:
### Key Features:
1. **Domain:**
The domain of \( f(x) \) includes all real numbers except where the denominator equals zero, as division by zero is undefined.
To find these points, solve \( x^2 - x = 0 \):
\[
x(x - 1) = 0 \implies x = 0 \text{ or } x = 1
\]
Therefore, the domain of \( f(x) \) is \( x \in \mathbb{R} \setminus \{0, 1\} \).
2. **Vertical Asymptotes:**
Vertical asymptotes occur where the denominator is zero but the numerator is not zero. From the previous step, since \( (2x - 5)^2 \neq 0 \) when \( x = 0 \) and \( x = 1 \), vertical asymptotes are at \( x = 0 \) and \( x = 1 \).
3. **Horizontal Asymptote:**
To determine the horizontal asymptote, consider the degrees of the polynomial in the numerator and the denominator. Both the numerator and the denominator are quadratic (degree 2). For such cases, compare the leading coefficients of the highest degree terms:
\[
f(x) \sim \frac{4x^2}{x^2} = 4 \quad \text{as} \quad x \to \pm \infty
\]
Hence, the horizontal asymptote is \( y = 4 \).
4. **Zeros:**
The zeros of the function occur where the numerator is zero. Solve \((2x - 5)^2 = 0\):
\[
2x - 5 = 0 \implies x = \frac{5}{2}
\]
Therefore,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5cd28f0e-7fb9-4455-a4c1-329c1161a16b%2Fdc71081b-b679-43d4-b9ba-1cb713799181%2Fzruyx7n_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 53: Rational Function Analysis**
Given the rational function \( f(x) = \frac{(2x - 5)^2}{x^2 - x} \), we are to analyze its behavior and properties.
**Function Description:**
- **Numerator:** \((2x - 5)^2\)
- **Denominator:** \(x^2 - x\)
This function can be broken down into the product and division of polynomial functions. Let's delve deeper into its characteristics:
### Key Features:
1. **Domain:**
The domain of \( f(x) \) includes all real numbers except where the denominator equals zero, as division by zero is undefined.
To find these points, solve \( x^2 - x = 0 \):
\[
x(x - 1) = 0 \implies x = 0 \text{ or } x = 1
\]
Therefore, the domain of \( f(x) \) is \( x \in \mathbb{R} \setminus \{0, 1\} \).
2. **Vertical Asymptotes:**
Vertical asymptotes occur where the denominator is zero but the numerator is not zero. From the previous step, since \( (2x - 5)^2 \neq 0 \) when \( x = 0 \) and \( x = 1 \), vertical asymptotes are at \( x = 0 \) and \( x = 1 \).
3. **Horizontal Asymptote:**
To determine the horizontal asymptote, consider the degrees of the polynomial in the numerator and the denominator. Both the numerator and the denominator are quadratic (degree 2). For such cases, compare the leading coefficients of the highest degree terms:
\[
f(x) \sim \frac{4x^2}{x^2} = 4 \quad \text{as} \quad x \to \pm \infty
\]
Hence, the horizontal asymptote is \( y = 4 \).
4. **Zeros:**
The zeros of the function occur where the numerator is zero. Solve \((2x - 5)^2 = 0\):
\[
2x - 5 = 0 \implies x = \frac{5}{2}
\]
Therefore,
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