Given $y=\frac{x}{x-1}$ And $x\ \textgreater \ 1$, Which Of The Following Is A Possible Value Of $ Y Y Y [/tex]?A. -1.9 B. -0.9 C. 0.0 D. 0.9 E. 1.9
Introduction
When dealing with rational functions, it's essential to understand the behavior of the function as the input variable changes. In this case, we're given the function $y=\frac{x}{x-1}$ and the constraint that $x > 1$. Our goal is to determine which of the given values is a possible value of $y$.
Understanding the Function
The given function is a rational function, which means it's the ratio of two polynomials. In this case, the numerator is $x$, and the denominator is $x-1$. Since $x > 1$, the denominator is always positive, which means the function is defined for all values of $x$ in the given domain.
Analyzing the Function
To understand the behavior of the function, let's analyze its components. The numerator $x$ is a linear function that increases as $x$ increases. The denominator $x-1$ is also a linear function, but it decreases as $x$ increases. Since the denominator is always positive, the function will increase as $x$ increases.
Finding the Possible Values of y
To find the possible values of $y$, we need to consider the range of the function. Since the function is defined for all values of $x$ in the given domain, we can find the range by analyzing the behavior of the function as $x$ approaches positive infinity and negative infinity.
Asymptotic Behavior
As $x$ approaches positive infinity, the function approaches 1. This is because the numerator $x$ approaches infinity, and the denominator $x-1$ approaches infinity as well. However, the ratio of the numerator to the denominator approaches 1.
Asymptotic Behavior (continued)
As $x$ approaches negative infinity, the function approaches 0. This is because the numerator $x$ approaches negative infinity, and the denominator $x-1$ approaches negative infinity as well. However, the ratio of the numerator to the denominator approaches 0.
Possible Values of y
Based on the asymptotic behavior of the function, we can conclude that the possible values of $y$ are all real numbers greater than 0. This is because the function approaches 1 as $x$ approaches positive infinity, and it approaches 0 as $x$ approaches negative infinity.
Checking the Options
Now that we know the possible values of $y$, let's check the options:
- A. -1.9: This value is not possible because $y$ must be greater than 0.
- B. -0.9: This value is not possible because $y$ must be greater than 0.
- C. 0.0: This value is not possible because $y$ must be greater than 0.
- D. 0.9: This value is possible because $y$ can be any real number greater than 0.
- E. 1.9: This value is possible because $y$ can be any real number greater than 0.
Conclusion
Based on the analysis of the function and its asymptotic behavior, we can conclude that the possible values of $y$ are all real numbers greater than 0. Therefore, the correct answer is D. 0.9.
Additional Discussion
It's worth noting that the function $y=\frac{x}{x-1}$ has a vertical asymptote at $x=1$, which means that the function is not defined at $x=1$. However, since $x > 1$, the function is defined for all values of $x$ in the given domain.
Final Thoughts
In conclusion, the possible values of $y$ in the function $y=\frac{x}{x-1}$ are all real numbers greater than 0. This is because the function approaches 1 as $x$ approaches positive infinity, and it approaches 0 as $x$ approaches negative infinity. Therefore, the correct answer is D. 0.9.
Introduction
In our previous article, we explored the possible values of $y$ in the rational function $y=\frac{x}{x-1}$, given the constraint that $x > 1$. We concluded that the possible values of $y$ are all real numbers greater than 0. In this article, we'll answer some frequently asked questions related to this topic.
Q: What is the domain of the function $y=\frac{x}{x-1}$?
A: The domain of the function is all real numbers greater than 1, since the denominator cannot be equal to 0.
Q: What is the range of the function $y=\frac{x}{x-1}$?
A: The range of the function is all real numbers greater than 0, since the function approaches 1 as $x$ approaches positive infinity and it approaches 0 as $x$ approaches negative infinity.
Q: Is the function $y=\frac{x}{x-1}$ continuous?
A: Yes, the function is continuous for all values of $x$ in the given domain, since the denominator is always positive.
Q: Does the function $y=\frac{x}{x-1}$ have any asymptotes?
A: Yes, the function has a vertical asymptote at $x=1$, since the denominator is equal to 0 at this point.
Q: Can the function $y=\frac{x}{x-1}$ be simplified?
A: Yes, the function can be simplified by multiplying both the numerator and the denominator by $x+1$, which results in $y=\frac{x(x+1)}{(x-1)(x+1)}=\frac{x(x+1)}{x^2-1}$.
Q: What is the behavior of the function $y=\frac{x}{x-1}$ as $x$ approaches positive infinity?
A: As $x$ approaches positive infinity, the function approaches 1.
Q: What is the behavior of the function $y=\frac{x}{x-1}$ as $x$ approaches negative infinity?
A: As $x$ approaches negative infinity, the function approaches 0.
Q: Can the function $y=\frac{x}{x-1}$ be used to model real-world phenomena?
A: Yes, the function can be used to model real-world phenomena, such as the growth of a population or the behavior of a physical system.
Q: How can the function $y=\frac{x}{x-1}$ be used in optimization problems?
A: The function can be used in optimization problems, such as finding the maximum or minimum value of a function subject to certain constraints.
Q: Can the function $y=\frac{x}{x-1}$ be used in machine learning?
A: Yes, the function can be used in machine learning, such as in the design of neural networks or the development of algorithms for classification and regression tasks.
Conclusion
In this article, we've answered some frequently asked questions related to the possible values of $y$ in the rational function $y=\frac{x}{x-1}$. We've discussed the domain and range of the function, its continuity and asymptotes, and its behavior as $x$ approaches positive and negative infinity. We've also explored the potential applications of the function in real-world phenomena, optimization problems, and machine learning.
Additional Discussion
It's worth noting that the function $y=\frac{x}{x-1}$ is a simple example of a rational function, and it can be used to model a wide range of real-world phenomena. By understanding the behavior of this function, we can gain insights into the behavior of more complex functions and systems.
Final Thoughts
In conclusion, the function $y=\frac{x}{x-1}$ is a powerful tool for modeling and analyzing real-world phenomena. By understanding its behavior and potential applications, we can gain a deeper understanding of the world around us and develop new solutions to complex problems.