Consider The Function $d(x)=\frac{x-2}{2x-6}$.A) Identify The Vertical Asymptote. Provide Work Or Justification For Your Answer.B) Identify The Horizontal Asymptote. Justify Your Answer.C) $d(x)=0$ When $x=$ ?

Introduction

In this article, we will explore the properties of a rational function, specifically the vertical and horizontal asymptotes, and the zeros of the function. We will use the function $d(x)=\frac{x-2}{2x-6}$ as a case study.

Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, and the numerator is not equal to zero.

To find the vertical asymptote of the function $d(x)=\frac{x-2}{2x-6}$, we need to find the value of $x$ that makes the denominator equal to zero.

2x - 6 = 0

Solving for $x$, we get:

2x = 6
x = 3

Therefore, the vertical asymptote of the function $d(x)=\frac{x-2}{2x-6}$ is $x=3$.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ approaches infinity or negative infinity. It occurs when the degree of the numerator is equal to the degree of the denominator.

To find the horizontal asymptote of the function $d(x)=\frac{x-2}{2x-6}$, we need to compare the degrees of the numerator and the denominator.

The degree of the numerator is 1, and the degree of the denominator is 1. Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

\frac{1}{2} = 0.5

Therefore, the horizontal asymptote of the function $d(x)=\frac{x-2}{2x-6}$ is $y=0.5$.

Zeros of the Function

A zero of a function is a value of $x$ that makes the function equal to zero. To find the zeros of the function $d(x)=\frac{x-2}{2x-6}$, we need to set the function equal to zero and solve for $x$.

\frac{x-2}{2x-6} = 0

Since the numerator is equal to zero, we can set $x-2=0$ and solve for $x$.

x - 2 = 0
x = 2

Therefore, the zero of the function $d(x)=\frac{x-2}{2x-6}$ is $x=2$.

Conclusion

In this article, we have explored the properties of a rational function, specifically the vertical and horizontal asymptotes, and the zeros of the function. We have used the function $d(x)=\frac{x-2}{2x-6}$ as a case study and found that the vertical asymptote is $x=3$, the horizontal asymptote is $y=0.5$, and the zero of the function is $x=2$.

References

• [1] "Rational Functions" by Math Open Reference
• [2] "Vertical and Horizontal Asymptotes" by Purplemath
• [3] "Zeros of a Function" by Math Is Fun

Further Reading

• "Rational Functions: Vertical and Horizontal Asymptotes" by Khan Academy
• "Zeros of a Rational Function" by Wolfram Alpha
• "Asymptotes and Zeros of a Rational Function" by MIT OpenCourseWare
  Q&A: Rational Functions, Vertical and Horizontal Asymptotes, and Zeros ====================================================================

Introduction

In our previous article, we explored the properties of a rational function, specifically the vertical and horizontal asymptotes, and the zeros of the function. In this article, we will answer some frequently asked questions related to rational functions, vertical and horizontal asymptotes, and zeros.

Q: What is a rational function?

A rational function is a function that can be written in the form of a fraction, where the numerator and denominator are polynomials. For example, the function $d(x)=\frac{x-2}{2x-6}$ is a rational function.

Q: What is a vertical asymptote?

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. It occurs when the denominator of a rational function is equal to zero, and the numerator is not equal to zero.

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

To find the vertical asymptote of a rational function, you need to find the value of $x$ that makes the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for $x$.

Q: What is a horizontal asymptote?

A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ approaches infinity or negative infinity. It occurs when the degree of the numerator is equal to the degree of the denominator.

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

To find the horizontal asymptote of a rational function, you need to compare the degrees of the numerator and the denominator. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

Q: What is a zero of a function?

A zero of a function is a value of $x$ that makes the function equal to zero. To find the zeros of a function, you need to set the function equal to zero and solve for $x$.

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

To find the zeros of a rational function, you need to set the numerator equal to zero and solve for $x$.

Q: Can a rational function have more than one vertical asymptote?

Yes, a rational function can have more than one vertical asymptote. This occurs when the denominator of the function is equal to zero at more than one value of $x$.

Q: Can a rational function have more than one horizontal asymptote?

No, a rational function can have only one horizontal asymptote. This occurs when the degree of the numerator is equal to the degree of the denominator.

Q: Can a rational function have a zero and a vertical asymptote at the same value of $x$?

Yes, a rational function can have a zero and a vertical asymptote at the same value of $x$. This occurs when the numerator and denominator of the function are both equal to zero at the same value of $x$.

Conclusion

In this article, we have answered some frequently asked questions related to rational functions, vertical and horizontal asymptotes, and zeros. We hope that this article has been helpful in clarifying some of the concepts related to rational functions.

References

• [1] "Rational Functions" by Math Open Reference
• [2] "Vertical and Horizontal Asymptotes" by Purplemath
• [3] "Zeros of a Function" by Math Is Fun

Further Reading

• "Rational Functions: Vertical and Horizontal Asymptotes" by Khan Academy
• "Zeros of a Rational Function" by Wolfram Alpha
• "Asymptotes and Zeros of a Rational Function" by MIT OpenCourseWare