Select The Correct Answer.Which Statement Describes The Graph Of The Function F ( X ) = X 2 − 1 X 2 − 2 X + 1 F(x)=\frac{x^2-1}{x^2-2x+1} F ( X ) = X 2 − 2 X + 1 X 2 − 1 ​ ?A. There Is A Hole At X = − 1 X=-1 X = − 1 .B. There Is A Vertical Asymptote At X = − 1 X=-1 X = − 1 .C. The Y Y Y -intercept Is

Introduction

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In this article, we will analyze the graph of the function $f(x)=\frac{x^2-1}{x^2-2x+1}$ and determine which statement describes its behavior.

The Function

The given function is $f(x)=\frac{x^2-1}{x^2-2x+1}$. To understand its behavior, we need to factor the numerator and denominator.

import sympy as sp
x = sp.symbols('x')
numerator = x2 - 1
denominator = x2 - 2*x + 1
factored_numerator = sp.factor(numerator)
factored_denominator = sp.factor(denominator)
print(factored_numerator)
print(factored_denominator)

The factored form of the numerator is $(x-1)(x+1)$, and the factored form of the denominator is $(x-1)^2$.

Holes and Vertical Asymptotes

A hole in a graph occurs when there is a factor in the numerator that cancels out a factor in the denominator. In this case, the factor $(x-1)$ appears in both the numerator and denominator, so there is a hole at $x=1$.

On the other hand, a vertical asymptote occurs when there is a factor in the denominator that does not cancel out with a factor in the numerator. In this case, the factor $(x-1)^2$ appears in the denominator, but there is no corresponding factor in the numerator. However, we need to consider the behavior of the function as $x$ approaches $-1$.

Behavior as $x$ Approaches $-1$

To determine the behavior of the function as $x$ approaches $-1$, we can use the factored form of the function:

$$f(x)=\frac{(x-1)(x+1)}{(x-1)^2}$$

As $x$ approaches $-1$, the numerator approaches $0$, and the denominator approaches $0$ as well. However, the denominator is a perfect square, so it approaches $0$ more quickly than the numerator. Therefore, the function approaches $\infty$ as $x$ approaches $-1$.

Conclusion

Based on our analysis, we can conclude that the graph of the function $f(x)=\frac{x^2-1}{x^2-2x+1}$ has a hole at $x=1$ and a vertical asymptote at $x=-1$.

Answer

The correct answer is:

B. There is a vertical asymptote at $x=-1$.

Discussion

This problem requires a deep understanding of rational functions and their behavior. The student needs to be able to factor the numerator and denominator, identify holes and vertical asymptotes, and analyze the behavior of the function as $x$ approaches certain values.

Additional Examples

1. Analyze the graph of the function $f(x)=\frac{x^2+1}{x^2-1}$.
2. Determine the behavior of the function $f(x)=\frac{x^2-4}{x^2-2x+1}$ as $x$ approaches $-1$.
3. Find the $y$-intercept of the function $f(x)=\frac{x^2-1}{x^2-2x+1}$.

Solutions

1. The graph of the function $f(x)=\frac{x^2+1}{x^2-1}$ has a hole at $x=1$ and a vertical asymptote at $x=-1$.
2. The function $f(x)=\frac{x^2-4}{x^2-2x+1}$ has a hole at $x=1$ and a vertical asymptote at $x=-1$.
3. The $y$-intercept of the function $f(x)=\frac{x^2-1}{x^2-2x+1}$ is $1$.
   Q&A: Rational Functions ==========================

Introduction

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In this article, we will answer some common questions about rational functions and provide examples to illustrate the concepts.

Q: What is a rational function?

A rational function is a function that can be expressed as the ratio of two polynomials. It is written in the form:

$$f(x)=\frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials.

Q: What is the difference between a rational function and a polynomial?

A polynomial is a function that can be expressed as the sum of terms, each of which is a constant times a power of $x$. A rational function, on the other hand, is a function that can be expressed as the ratio of two polynomials.

Q: How do I factor a rational function?

To factor a rational function, you need to factor the numerator and denominator separately. The numerator and denominator can be factored using the same techniques as for polynomials.

Q: What is a hole in a rational function?

A hole in a rational function occurs when there is a factor in the numerator that cancels out a factor in the denominator. This means that the function is not defined at that point, but it is still continuous.

Q: What is a vertical asymptote in a rational function?

A vertical asymptote in a rational function occurs when there is a factor in the denominator that does not cancel out with a factor in the numerator. This means that the function approaches infinity as $x$ approaches that point.

Q: How do I find the $y$-intercept of a rational function?

To find the $y$-intercept of a rational function, you need to substitute $x=0$ into the function and simplify.

Q: Can a rational function have a horizontal asymptote?

Yes, a rational function can have a horizontal asymptote. This occurs when the degree of the numerator is less than or equal to the degree of the denominator.

Q: Can a rational function have a slant asymptote?

Yes, a rational function can have a slant asymptote. This occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Q: How do I graph a rational function?

To graph a rational function, you need to identify the $x$-intercepts, $y$-intercepts, holes, and vertical asymptotes. You can then use this information to sketch the graph of the function.

Q: Can a rational function be a quadratic function?

Yes, a rational function can be a quadratic function. This occurs when the numerator and denominator are both quadratic polynomials.

Q: Can a rational function be a linear function?

Yes, a rational function can be a linear function. This occurs when the numerator and denominator are both linear polynomials.

Q: Can a rational function be a constant function?

Yes, a rational function can be a constant function. This occurs when the numerator and denominator are both constant polynomials.

Conclusion

Rational functions are a type of function that can be expressed as the ratio of two polynomials. They can have holes, vertical asymptotes, and horizontal or slant asymptotes. By understanding the properties of rational functions, you can graph them and solve equations involving them.

Additional Examples

1. Analyze the graph of the function $f(x)=\frac{x^2+1}{x^2-1}$.
2. Determine the behavior of the function $f(x)=\frac{x^2-4}{x^2-2x+1}$ as $x$ approaches $-1$.
3. Find the $y$-intercept of the function $f(x)=\frac{x^2-1}{x^2-2x+1}$.

Solutions

1. The graph of the function $f(x)=\frac{x^2+1}{x^2-1}$ has a hole at $x=1$ and a vertical asymptote at $x=-1$.
2. The function $f(x)=\frac{x^2-4}{x^2-2x+1}$ has a hole at $x=1$ and a vertical asymptote at $x=-1$.
3. The $y$-intercept of the function $f(x)=\frac{x^2-1}{x^2-2x+1}$ is $1$.