What Is The Horizontal Asymptote Of $f(x)=\frac{-2x}{x+1}$?A. $y=-2$ B. $y=-1$ C. $y=0$ D. $y=1$

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Understanding Horizontal Asymptotes

In mathematics, a horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. In other words, it's the horizontal line that the function gets arbitrarily close to as x goes to positive or negative infinity. Horizontal asymptotes are an essential concept in calculus and are used to analyze the behavior of functions.

Horizontal Asymptotes of Rational Functions

Rational functions are functions that can be written as the ratio of two polynomials. The general form of a rational function is f(x) = p(x)/q(x), where p(x) and q(x) are polynomials. To find the horizontal asymptote of a rational function, we need to examine the degrees of the polynomials in the numerator and denominator.

Degree of the Numerator and Denominator

The degree of a polynomial is the highest power of the variable (in this case, x) in the polynomial. For example, the polynomial 3x^2 + 2x + 1 has a degree of 2, while the polynomial x^3 - 2x^2 + x - 1 has a degree of 3.

Finding the Horizontal Asymptote

To find the horizontal asymptote of a rational function, we need to compare the degrees of the polynomials in the numerator and denominator. There are three cases to consider:

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
  • If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Example: Finding the Horizontal Asymptote of a Rational Function

Let's consider the rational function f(x) = -2x / (x + 1). To find the horizontal asymptote, we need to compare the degrees of the polynomials in the numerator and denominator.

The numerator is -2x, which has a degree of 1. The denominator is x + 1, which also has a degree of 1. Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.

The leading coefficient of the numerator is -2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = -2 / 1 = -2.

Conclusion

In conclusion, the horizontal asymptote of a rational function can be found by comparing the degrees of the polynomials in the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Answer

The horizontal asymptote of the rational function f(x) = -2x / (x + 1) is y = -2.

Final Answer

The final answer is A\boxed{A}

Understanding Horizontal Asymptotes

In mathematics, a horizontal asymptote is a horizontal line that a function approaches as the absolute value of the x-coordinate gets larger and larger. In other words, it's the horizontal line that the function gets arbitrarily close to as x goes to positive or negative infinity. Horizontal asymptotes are an essential concept in calculus and are used to analyze the behavior of functions.

Q&A: Horizontal Asymptotes of Rational Functions

Q: What is the horizontal asymptote of a rational function?

A: The horizontal asymptote of a rational function is a horizontal line that the function approaches as the absolute value of the x-coordinate gets larger and larger.

Q: How do I find the horizontal asymptote of a rational function?

A: To find the horizontal asymptote of a rational function, you need to compare the degrees of the polynomials in the numerator and denominator. There are three cases to consider:

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
  • If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Q: What is the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable (in this case, x) in the polynomial. For example, the polynomial 3x^2 + 2x + 1 has a degree of 2, while the polynomial x^3 - 2x^2 + x - 1 has a degree of 3.

Q: How do I find the leading coefficient of a polynomial?

A: The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. For example, the polynomial 3x^2 + 2x + 1 has a leading coefficient of 3, while the polynomial x^3 - 2x^2 + x - 1 has a leading coefficient of 1.

Q: What is the horizontal asymptote of the rational function f(x) = -2x / (x + 1)?

A: To find the horizontal asymptote of the rational function f(x) = -2x / (x + 1), we need to compare the degrees of the polynomials in the numerator and denominator. The numerator is -2x, which has a degree of 1. The denominator is x + 1, which also has a degree of 1. Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.

The leading coefficient of the numerator is -2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = -2 / 1 = -2.

Q: What is the horizontal asymptote of the rational function f(x) = x^2 / (x + 1)?

A: To find the horizontal asymptote of the rational function f(x) = x^2 / (x + 1), we need to compare the degrees of the polynomials in the numerator and denominator. The numerator is x^2, which has a degree of 2. The denominator is x + 1, which has a degree of 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Q: What is the horizontal asymptote of the rational function f(x) = 2 / (x^2 + 1)?

A: To find the horizontal asymptote of the rational function f(x) = 2 / (x^2 + 1), we need to compare the degrees of the polynomials in the numerator and denominator. The numerator is 2, which has a degree of 0. The denominator is x^2 + 1, which has a degree of 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Conclusion

In conclusion, the horizontal asymptote of a rational function can be found by comparing the degrees of the polynomials in the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Final Answer

The final answer is A\boxed{A}