Find The Vertical Asymptotes, If Any, Of The Graph Of The Rational Function.${ G(x) = \frac{x}{x^2 - 25} }$A. { X = 5 $}$B. { X = 5, X = -5 $}$C. { X = 5, X = -5, X = 0 $}$D. No Vertical Asymptote
Introduction
In mathematics, a rational function is a function that can be expressed as the ratio of two polynomials. Rational functions are commonly used in algebra, calculus, and other branches of mathematics to model real-world phenomena. One of the key concepts in understanding the behavior of rational functions is the concept of vertical asymptotes. In this article, we will explore how to find the vertical asymptotes of a rational function.
What are Vertical Asymptotes?
A vertical asymptote is a vertical line that a rational function approaches but never touches. In other words, it is a line that the function gets arbitrarily close to but never crosses. Vertical asymptotes are an important concept in understanding the behavior of rational functions, as they can help us identify the points where the function is undefined.
Finding Vertical Asymptotes
To find the vertical asymptotes of a rational function, we need to look for the values of x that make the denominator of the function equal to zero. This is because a rational function is undefined when the denominator is equal to zero. Therefore, we need to find the values of x that make the denominator equal to zero and exclude them from the domain of the function.
Step 1: Factor the Denominator
The first step in finding the vertical asymptotes of a rational function is to factor the denominator. In this case, the denominator is x^2 - 25, which can be factored as (x + 5)(x - 5).
Step 2: Set the Denominator Equal to Zero
Next, we need to set the denominator equal to zero and solve for x. In this case, we have (x + 5)(x - 5) = 0. We can solve for x by setting each factor equal to zero and solving for x.
Solving for x
Setting the first factor equal to zero, we get x + 5 = 0, which gives us x = -5. Setting the second factor equal to zero, we get x - 5 = 0, which gives us x = 5.
Conclusion
Therefore, the vertical asymptotes of the rational function g(x) = x / (x^2 - 25) are x = 5 and x = -5. These are the values of x that make the denominator equal to zero and exclude them from the domain of the function.
Answer
The correct answer is B. x = 5, x = -5.
Discussion
In this article, we have explored how to find the vertical asymptotes of a rational function. We have seen that the vertical asymptotes are the values of x that make the denominator equal to zero and exclude them from the domain of the function. We have also seen that the vertical asymptotes can be found by factoring the denominator and setting it equal to zero.
Example Problems
Here are some example problems to help you practice finding vertical asymptotes:
- Find the vertical asymptotes of the rational function f(x) = x / (x^2 - 4).
- Find the vertical asymptotes of the rational function g(x) = x / (x^2 - 9).
- Find the vertical asymptotes of the rational function h(x) = x / (x^2 - 16).
Solutions
- The vertical asymptotes of the rational function f(x) = x / (x^2 - 4) are x = 2 and x = -2.
- The vertical asymptotes of the rational function g(x) = x / (x^2 - 9) are x = 3 and x = -3.
- The vertical asymptotes of the rational function h(x) = x / (x^2 - 16) are x = 4 and x = -4.
Conclusion
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about vertical asymptotes.
Q: What is a vertical asymptote?
A: A vertical asymptote is a vertical line that a rational function approaches but never touches. In other words, it is a line that the function gets arbitrarily close to but never crosses.
Q: How do I find the vertical asymptotes of a rational function?
A: To find the vertical asymptotes of a rational function, you need to look for the values of x that make the denominator of the function equal to zero. This is because a rational function is undefined when the denominator is equal to zero.
Q: What is the difference between a vertical asymptote and a hole in a graph?
A: A vertical asymptote is a vertical line that a rational function approaches but never touches. A hole in a graph, on the other hand, is a point where the function is undefined but the graph passes through that point. In other words, a hole is a point where the function is not defined but the graph is still continuous.
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 more than one factor that equals zero.
Q: How do I determine if a rational function has a vertical asymptote at a particular point?
A: To determine if a rational function has a vertical asymptote at a particular point, you need to check if the denominator of the function equals zero at that point. If the denominator equals zero, then the function has a vertical asymptote at that point.
Q: Can a rational function have a vertical asymptote at x = 0?
A: Yes, a rational function can have a vertical asymptote at x = 0. This occurs when the denominator of the function has a factor of x that equals zero.
Q: How do I find the vertical asymptotes of a rational function with a quadratic denominator?
A: To find the vertical asymptotes of a rational function with a quadratic denominator, you need to factor the denominator and set it equal to zero. Then, solve for x to find the vertical asymptotes.
Q: Can a rational function have a vertical asymptote at a point where the numerator and denominator are both zero?
A: No, a rational function cannot have a vertical asymptote at a point where the numerator and denominator are both zero. This is because the function is undefined at that point, but it is not a vertical asymptote.
Q: How do I determine if a rational function has a vertical asymptote at a point where the denominator is a perfect square?
A: To determine if a rational function has a vertical asymptote at a point where the denominator is a perfect square, you need to check if the denominator equals zero at that point. If the denominator equals zero, then the function has a vertical asymptote at that point.
Q: Can a rational function have a vertical asymptote at a point where the denominator is a difference of squares?
A: Yes, a rational function can have a vertical asymptote at a point where the denominator is a difference of squares. This occurs when the denominator can be factored into two binomials that equal zero.
Conclusion
In conclusion, vertical asymptotes are an important concept in understanding the behavior of rational functions. By following the steps outlined in this article, you can find the vertical asymptotes of any rational function. Remember to factor the denominator, set it equal to zero, and solve for x to find the vertical asymptotes.