At Which Values Of $x$ Does The Graph Of The Function $f(x)$ Have A Vertical Asymptote? Check All That Apply. F ( X ) = X − 5 X 2 + 3 X − 18 F(x) = \frac{x-5}{x^2+3x-18} F ( X ) = X 2 + 3 X − 18 X − 5 A. X = 5 X=5 X = 5 B. X = 6 X=6 X = 6 C. X = 3 X=3 X = 3 D. X = − 5 X=-5 X = − 5 E.
Introduction
In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the graph of the function gets arbitrarily close to, but never crosses. In the context of rational functions, vertical asymptotes occur when the denominator of the function is equal to zero, and the numerator is not equal to zero. In this article, we will explore the concept of vertical asymptotes and determine the values of x for which the graph of the function f(x) = (x-5)/(x^2+3x-18) has a vertical asymptote.
What are Vertical Asymptotes?
A vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the graph of the function gets arbitrarily close to, but never crosses. Vertical asymptotes occur when the denominator of a rational function is equal to zero, and the numerator is not equal to zero. In other words, a vertical asymptote occurs when the function is undefined at a particular value of x.
How to Find Vertical Asymptotes
To find the vertical asymptotes of a rational function, we need to follow these steps:
- Factor the denominator of the function.
- Set the denominator equal to zero and solve for x.
- Check if the numerator is equal to zero at the same value of x.
- If the numerator is not equal to zero, then the function has a vertical asymptote at that value of x.
Finding Vertical Asymptotes for f(x) = (x-5)/(x^2+3x-18)
To find the vertical asymptotes of the function f(x) = (x-5)/(x^2+3x-18), we need to follow the steps outlined above.
Step 1: Factor the Denominator
The denominator of the function is x^2+3x-18. We can factor this expression as (x+6)(x-3).
Step 2: Set the Denominator Equal to Zero and Solve for x
We set the denominator equal to zero and solve for x: (x+6)(x-3) = 0. This gives us two possible values for x: x = -6 and x = 3.
Step 3: Check if the Numerator is Equal to Zero at the Same Value of x
We need to check if the numerator is equal to zero at the same value of x. The numerator is x-5. We substitute x = -6 and x = 3 into the numerator and check if it is equal to zero.
For x = -6, the numerator is -6-5 = -11, which is not equal to zero.
For x = 3, the numerator is 3-5 = -2, which is not equal to zero.
Step 4: Determine the Vertical Asymptotes
Since the numerator is not equal to zero at x = -6 and x = 3, we conclude that the function has vertical asymptotes at x = -6 and x = 3.
Conclusion
In this article, we explored the concept of vertical asymptotes and determined the values of x for which the graph of the function f(x) = (x-5)/(x^2+3x-18) has a vertical asymptote. We followed the steps outlined above to find the vertical asymptotes of the function and concluded that the function has vertical asymptotes at x = -6 and x = 3.
Answer
The correct answers are:
- C.
- B. is not correct because the denominator is zero at x = 6 but the numerator is also zero at x = 6, so it is not a vertical asymptote.
- A. is not correct because the denominator is not zero at x = 5.
- D. is not correct because the denominator is not zero at x = -5.
Therefore, the correct answer is:
- C.
- B.
Vertical Asymptotes of Rational Functions: Q&A =====================================================
Introduction
In our previous article, we explored the concept of vertical asymptotes and determined the values of x for which the graph of the function f(x) = (x-5)/(x^2+3x-18) has a vertical asymptote. In this article, we will answer some frequently asked questions about vertical asymptotes and provide additional examples to help solidify your understanding of this concept.
Q&A
Q: What is a vertical asymptote?
A: A vertical asymptote is a vertical line that a function approaches but never touches. It is a line that the graph of the function gets arbitrarily close to, but never crosses.
Q: How do you find vertical asymptotes?
A: To find the vertical asymptotes of a rational function, you need to follow these steps:
- Factor the denominator of the function.
- Set the denominator equal to zero and solve for x.
- Check if the numerator is equal to zero at the same value of x.
- If the numerator is not equal to zero, then the function has a vertical asymptote at that value of x.
Q: What happens if the numerator and denominator are both zero at the same value of x?
A: If the numerator and denominator are both zero at the same value of x, then the function has a hole at that value of x, not a vertical asymptote.
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 of the function has multiple factors that are equal to zero.
Q: How do you determine the equation of a vertical asymptote?
A: The equation of a vertical asymptote is a vertical line that passes through the value of x where the denominator is equal to zero. The equation of the vertical asymptote is x = a, where a is the value of x where the denominator is equal to zero.
Q: Can a function have a vertical asymptote at x = 0?
A: Yes, a function can have a vertical asymptote at x = 0. This occurs when the denominator of the function is equal to zero at x = 0.
Q: Can a function have a vertical asymptote at x = infinity?
A: No, a function cannot have a vertical asymptote at x = infinity. Vertical asymptotes occur at finite values of x.
Examples
Example 1: Finding Vertical Asymptotes
Find the vertical asymptotes of the function f(x) = (x-2)/(x^2-4x+3).
Step 1: Factor the Denominator
The denominator of the function is x^2-4x+3. We can factor this expression as (x-3)(x-1).
Step 2: Set the Denominator Equal to Zero and Solve for x
We set the denominator equal to zero and solve for x: (x-3)(x-1) = 0. This gives us two possible values for x: x = 3 and x = 1.
Step 3: Check if the Numerator is Equal to Zero at the Same Value of x
We need to check if the numerator is equal to zero at the same value of x. The numerator is x-2. We substitute x = 3 and x = 1 into the numerator and check if it is equal to zero.
For x = 3, the numerator is 3-2 = 1, which is not equal to zero.
For x = 1, the numerator is 1-2 = -1, which is not equal to zero.
Step 4: Determine the Vertical Asymptotes
Since the numerator is not equal to zero at x = 3 and x = 1, we conclude that the function has vertical asymptotes at x = 3 and x = 1.
Example 2: Finding Vertical Asymptotes with a Hole
Find the vertical asymptotes of the function f(x) = (x-2)/(x^2-4x+4).
Step 1: Factor the Denominator
The denominator of the function is x^2-4x+4. We can factor this expression as (x-2)^2.
Step 2: Set the Denominator Equal to Zero and Solve for x
We set the denominator equal to zero and solve for x: (x-2)^2 = 0. This gives us one possible value for x: x = 2.
Step 3: Check if the Numerator is Equal to Zero at the Same Value of x
We need to check if the numerator is equal to zero at the same value of x. The numerator is x-2. We substitute x = 2 into the numerator and check if it is equal to zero.
For x = 2, the numerator is 2-2 = 0, which is equal to zero.
Step 4: Determine the Vertical Asymptotes
Since the numerator is equal to zero at x = 2, we conclude that the function has a hole at x = 2, not a vertical asymptote.
Conclusion
In this article, we answered some frequently asked questions about vertical asymptotes and provided additional examples to help solidify your understanding of this concept. We also explored the concept of holes in rational functions and how they differ from vertical asymptotes. By following the steps outlined in this article, you can determine the vertical asymptotes of a rational function and understand the behavior of the function at those points.