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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Question
Choose the correct slant asymptote for the function.
y = x + 2
y = x - 2
y = x + 3
y = x - 3
![## Rational Function Analysis
**Function Definition:**
\[ f(x) = \frac{x^2 - 5x + 6}{x - 2} \]
### Detailed Breakdown:
This expression represents a rational function, which is defined as the quotient of two polynomials. Let's analyze each component of the function:
**Numerator:** \( x^2 - 5x + 6 \)
This is a quadratic polynomial, and it can be factored as:
\[ x^2 - 5x + 6 = (x - 3)(x - 2) \]
**Denominator:** \( x - 2 \)
This is a linear polynomial.
When simplified, the function becomes:
\[ f(x) = \frac{(x - 3)(x - 2)}{x - 2} \]
Here, \( x - 2 \) cancels out in the numerator and the denominator, but we must remember that this operation leaves a hole in the graph at \( x = 2 \). Therefore, the simplified form is:
\[ f(x) = x - 3 \]
However, this simplified form \( f(x) = x - 3 \) is valid only for \( x \neq 2 \). At \( x = 2 \), the function \( f(x) \) is undefined.
### Graphical Interpretation:
The graph of \( f(x) = \frac{x^2 - 5x + 6}{x - 2} \) will have:
- A hole at \( x = 2 \).
- The remaining part of the graph will follow the linear equation \( y = x - 3 \).
### Key Points:
- **Hole:** The function is undefined at \( x = 2 \). There is a hole in the graph at this point.
- **Line:** Apart from the hole at \( x = 2 \), the graph follows the line \( y = x - 3 \).
Understanding how to analyze and graph rational functions is crucial in algebra and calculus. This example provides insight into handling undefined points and simplifying complex expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f7b3938-790b-4169-a858-04369df79473%2Fb77df9dd-16a3-4c00-a546-d955d2355af3%2Fthb5jif_processed.png&w=3840&q=75)
Transcribed Image Text:## Rational Function Analysis
**Function Definition:**
\[ f(x) = \frac{x^2 - 5x + 6}{x - 2} \]
### Detailed Breakdown:
This expression represents a rational function, which is defined as the quotient of two polynomials. Let's analyze each component of the function:
**Numerator:** \( x^2 - 5x + 6 \)
This is a quadratic polynomial, and it can be factored as:
\[ x^2 - 5x + 6 = (x - 3)(x - 2) \]
**Denominator:** \( x - 2 \)
This is a linear polynomial.
When simplified, the function becomes:
\[ f(x) = \frac{(x - 3)(x - 2)}{x - 2} \]
Here, \( x - 2 \) cancels out in the numerator and the denominator, but we must remember that this operation leaves a hole in the graph at \( x = 2 \). Therefore, the simplified form is:
\[ f(x) = x - 3 \]
However, this simplified form \( f(x) = x - 3 \) is valid only for \( x \neq 2 \). At \( x = 2 \), the function \( f(x) \) is undefined.
### Graphical Interpretation:
The graph of \( f(x) = \frac{x^2 - 5x + 6}{x - 2} \) will have:
- A hole at \( x = 2 \).
- The remaining part of the graph will follow the linear equation \( y = x - 3 \).
### Key Points:
- **Hole:** The function is undefined at \( x = 2 \). There is a hole in the graph at this point.
- **Line:** Apart from the hole at \( x = 2 \), the graph follows the line \( y = x - 3 \).
Understanding how to analyze and graph rational functions is crucial in algebra and calculus. This example provides insight into handling undefined points and simplifying complex expressions.
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