The Given Rational Function

Oct 13, 2025 by pynkocean.com 73 views

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Introduction

Rational functions are a type of function that can be expressed as the ratio of two polynomials. In this article, we will explore the properties of a given rational function, including finding the intercepts, determining the domain, and identifying any vertical or horizontal asymptotes.

The Given Rational Function

The given rational function is:

${ f(x) = \frac{3x}{x-6} }$

(A) Finding the Intercepts

To find the intercepts of the graph, we need to find the points where the graph intersects the x-axis and the y-axis.

Finding the x-Intercepts

The x-intercepts are the points where the graph intersects the x-axis, i.e., where y = 0. To find the x-intercepts, we set f(x) = 0 and solve for x.

${ \frac{3x}{x-6} = 0 }$

Since the numerator is equal to zero, we have:

${ 3x = 0 }$

Solving for x, we get:

${ x = 0 }$

Therefore, the x-intercept is (0, 0).

Finding the y-Intercept

The y-intercept is the point where the graph intersects the y-axis, i.e., where x = 0. To find the y-intercept, we substitute x = 0 into the function.

${ f(0) = \frac{3(0)}{0-6} }$

Simplifying, we get:

${ f(0) = \frac{0}{-6} = 0 }$

Therefore, the y-intercept is (0, 0).

(B) Determining the Domain

The domain of a function is the set of all possible input values for which the function is defined. For a rational function, the domain is all real numbers except for the values that make the denominator equal to zero.

In this case, the denominator is x - 6, which is equal to zero when x = 6. Therefore, the domain of the function is all real numbers except for x = 6.

(C) Finding Vertical and Horizontal Asymptotes

Vertical asymptotes are the vertical lines that the graph approaches as x approaches a certain value. Horizontal asymptotes are the horizontal lines that the graph approaches as x approaches infinity.

Finding Vertical Asymptotes

To find the vertical asymptotes, we need to find the values of x that make the denominator equal to zero. In this case, the denominator is x - 6, which is equal to zero when x = 6. Therefore, the vertical asymptote is x = 6.

Finding Horizontal Asymptotes

To find the horizontal asymptotes, we need to compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is the ratio of the leading coefficients.

In this case, the degree of the numerator is 1, and the degree of the denominator is 1. Therefore, the horizontal asymptote is the ratio of the leading coefficients, which is 3/1 = 3.

(D) Graphing the Function

To graph the function, we can use a graphing calculator or software. The graph of the function is a rational function with a vertical asymptote at x = 6 and a horizontal asymptote at y = 3.

Conclusion

In this article, we have explored the properties of a given rational function, including finding the intercepts, determining the domain, and identifying any vertical or horizontal asymptotes. We have also graphed the function using a graphing calculator or software.

Key Takeaways

The x-intercept of the graph is (0, 0).
The y-intercept of the graph is (0, 0).
The domain of the function is all real numbers except for x = 6.
The vertical asymptote is x = 6.
The horizontal asymptote is y = 3.

Final Thoughts

Rational functions are an important type of function in mathematics, and understanding their properties is essential for solving problems in algebra, calculus, and other areas of mathematics. By following the steps outlined in this article, you can find the intercepts, determine the domain, and identify any vertical or horizontal asymptotes of a rational function.

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Introduction

In our previous article, we explored the properties of a given rational function, including finding the intercepts, determining the domain, and identifying any vertical or horizontal asymptotes. In this article, we will answer some frequently asked questions about rational functions.

Q&A

Q: What is a rational function?

A: A rational function is a type of function that can be expressed as the ratio of two polynomials.

Q: What are the intercepts of a rational function?

A: The intercepts of a rational function are the points where the graph intersects the x-axis and the y-axis. To find the x-intercepts, we set f(x) = 0 and solve for x. To find the y-intercept, we substitute x = 0 into the function.

Q: How do I determine the domain of a rational function?

A: The domain of a rational function is all real numbers except for the values that make the denominator equal to zero. To find the domain, we need to find the values of x that make the denominator equal to zero and exclude them from the domain.

Q: What are vertical and horizontal asymptotes?

A: Vertical asymptotes are the vertical lines that the graph approaches as x approaches a certain value. Horizontal asymptotes are the horizontal lines that the graph approaches as x approaches infinity.

Q: How do I find vertical asymptotes?

A: To find vertical asymptotes, we need to find the values of x that make the denominator equal to zero. These values are the vertical asymptotes.

Q: How do I find horizontal asymptotes?

A: To find horizontal asymptotes, we need to compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is the ratio of the leading coefficients.

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

A: Yes, a rational function can have more than one vertical asymptote. This occurs when the denominator has more than one factor that equals zero.

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

A: No, a rational function can have at most one horizontal asymptote. This occurs when the degree of the numerator is less than the degree of the denominator.

Q: How do I graph a rational function?

A: To graph a rational function, we can use a graphing calculator or software. We can also use the properties of the function, such as the intercepts, domain, and asymptotes, to sketch the graph.

Q: What are some common mistakes to avoid when working with rational functions?

A: Some common mistakes to avoid when working with rational functions include:

Not simplifying the function before finding the domain or asymptotes.
Not checking for vertical asymptotes.
Not comparing the degrees of the numerator and denominator when finding horizontal asymptotes.
Not using a graphing calculator or software to check the graph.

Conclusion

In this article, we have answered some frequently asked questions about rational functions. We have covered topics such as intercepts, domain, vertical and horizontal asymptotes, and graphing. By following the steps outlined in this article, you can better understand rational functions and how to work with them.

Key Takeaways

A rational function is a type of function that can be expressed as the ratio of two polynomials.
The intercepts of a rational function are the points where the graph intersects the x-axis and the y-axis.
The domain of a rational function is all real numbers except for the values that make the denominator equal to zero.
Vertical asymptotes are the vertical lines that the graph approaches as x approaches a certain value.
Horizontal asymptotes are the horizontal lines that the graph approaches as x approaches infinity.

Final Thoughts

Rational functions are an important type of function in mathematics, and understanding their properties is essential for solving problems in algebra, calculus, and other areas of mathematics. By following the steps outlined in this article, you can better understand rational functions and how to work with them.