Let $f(x)=\frac{15}{1+4 E^{-0.2 X}}$What Are The Asymptotes Of The Graph Of $f(x)$?Select Each Correct Answer.A. Y = − 0.2 Y=-0.2 Y = − 0.2 B. Y = 4 Y=4 Y = 4 C. Y = 0 Y=0 Y = 0 D. Y = 15 Y=15 Y = 15

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Introduction

In mathematics, an asymptote is a line that a function approaches as the input (or independent variable) gets arbitrarily large or approaches a certain value. Asymptotes are an essential concept in understanding the behavior of functions, especially rational functions. In this article, we will explore the asymptotes of the graph of the function f(x)=151+4e−0.2xf(x)=\frac{15}{1+4 e^{-0.2 x}}.

What are Asymptotes?

Asymptotes can be vertical, horizontal, or oblique. A vertical asymptote occurs when a function approaches positive or negative infinity as the input gets arbitrarily close to a certain value. A horizontal asymptote occurs when a function approaches a constant value as the input gets arbitrarily large. An oblique asymptote occurs when a function approaches a linear function as the input gets arbitrarily large.

Horizontal Asymptotes

To find the horizontal asymptotes of a rational function, we need to compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0y=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. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Vertical Asymptotes

To find the vertical asymptotes of a rational function, we need to find the values of the input that make the denominator equal to zero. These values are called the vertical asymptotes.

Oblique Asymptotes

To find the oblique asymptotes of a rational function, we need to divide the numerator by the denominator using polynomial long division or synthetic division. The quotient is the oblique asymptote.

Asymptotes of the Graph of f(x)f(x)

Now, let's find the asymptotes of the graph of f(x)=151+4e−0.2xf(x)=\frac{15}{1+4 e^{-0.2 x}}. We can see that the degree of the numerator is 0 and the degree of the denominator is 1. Therefore, the horizontal asymptote is y=0y=0.

To find the vertical asymptotes, we need to find the values of the input that make the denominator equal to zero. The denominator is 1+4e−0.2x1+4 e^{-0.2 x}. We can see that the denominator is never equal to zero, so there are no vertical asymptotes.

To find the oblique asymptotes, we need to divide the numerator by the denominator using polynomial long division or synthetic division. However, since the degree of the numerator is 0, the quotient is a constant, and there is no oblique asymptote.

Conclusion

In conclusion, the asymptotes of the graph of f(x)=151+4e−0.2xf(x)=\frac{15}{1+4 e^{-0.2 x}} are:

  • Horizontal asymptote: y=0y=0
  • Vertical asymptote: None
  • Oblique asymptote: None

Therefore, the correct answers are:

A. y=−0.2y=-0.2 (Incorrect) B. y=4y=4 (Incorrect) C. y=0y=0 (Correct) D. y=15y=15 (Incorrect)

Discussion

What are your thoughts on asymptotes? Have you ever encountered a function with no asymptotes? Share your experiences and insights in the comments below!

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Algebra and Trigonometry, 4th edition, James Stewart
  • [3] Calculus: Early Transcendentals, 7th edition, James Stewart

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Introduction

In our previous article, we explored the asymptotes of the graph of the function f(x)=151+4e−0.2xf(x)=\frac{15}{1+4 e^{-0.2 x}}. We found that the horizontal asymptote is y=0y=0, and there are no vertical or oblique asymptotes. In this article, we will answer some frequently asked questions about asymptotes.

Q: What is the difference between a horizontal asymptote and a vertical asymptote?

A: A horizontal asymptote is a line that a function approaches as the input gets arbitrarily large. A vertical asymptote is a line that a function approaches as the input gets arbitrarily close to a certain value.

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 numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0y=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. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

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

A: To find the vertical asymptote of a rational function, you need to find the values of the input that make the denominator equal to zero. These values are called the vertical asymptotes.

Q: What is an oblique asymptote?

A: An oblique asymptote is a line that a function approaches as the input gets arbitrarily large. It is a linear function that the function approaches as the input gets arbitrarily large.

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

A: To find the oblique asymptote of a rational function, you need to divide the numerator by the denominator using polynomial long division or synthetic division. The quotient is the oblique asymptote.

Q: Can a function have no asymptotes?

A: Yes, a function can have no asymptotes. This occurs when the degree of the numerator is less than the degree of the denominator, and the leading coefficient of the numerator is zero.

Q: What are some common mistakes to avoid when finding asymptotes?

A: Some common mistakes to avoid when finding asymptotes include:

  • Not comparing the degrees of the numerator and denominator when finding the horizontal asymptote.
  • Not finding the values of the input that make the denominator equal to zero when finding the vertical asymptote.
  • Not using polynomial long division or synthetic division when finding the oblique asymptote.
  • Not checking if the function has any holes or gaps in its graph.

Conclusion

In conclusion, asymptotes are an essential concept in understanding the behavior of functions, especially rational functions. By following the steps outlined in this article, you can find the asymptotes of a rational function and gain a deeper understanding of its behavior.

Discussion

What are your thoughts on asymptotes? Have you ever encountered a function with no asymptotes? Share your experiences and insights in the comments below!

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Algebra and Trigonometry, 4th edition, James Stewart
  • [3] Calculus: Early Transcendentals, 7th edition, James Stewart

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