Choose the correct slant asymptote for the function.y = x + 2y = x - 2y = x + 3y = 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.

Answered: х -5х +6 f(x) = х - 2

Math

Algebra

х -5х +6 f(x) = х - 2

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.

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