Select The Correct Answer.Which Function Has All Of These Features?I. The Function Is Increasing. II. The Function Has A Domain Of $(1, \infty$\]. III. For Low Values Of $x$, The Values Of $f(x$\] Move Toward Negative

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Introduction

In mathematics, functions are used to describe the relationship between variables. They can be increasing, decreasing, or have a combination of both. In this article, we will discuss the characteristics of a function that has an increasing nature, a domain of (1,∞)(1, \infty), and values that move toward negative for low values of xx. We will analyze the given features and determine which function satisfies all these conditions.

Understanding the Features

I. The Function is Increasing

An increasing function is one where the output value increases as the input value increases. In other words, as xx increases, f(x)f(x) also increases. This is a fundamental property of functions that we will use to analyze the given features.

II. The Function has a Domain of (1,∞)(1, \infty)

The domain of a function is the set of all possible input values for which the function is defined. In this case, the domain is (1,∞)(1, \infty), which means that the function is defined for all real numbers greater than 1. This implies that the function is not defined for x≤1x \leq 1.

III. For Low Values of xx, the Values of f(x)f(x) Move Toward Negative

This feature implies that as xx approaches 1 from the right, f(x)f(x) approaches negative infinity. In other words, the function has a vertical asymptote at x=1x = 1 and approaches negative infinity as xx approaches 1 from the right.

Analyzing the Functions

Function 1: f(x)=1x−1f(x) = \frac{1}{x-1}

This function has a domain of (1,∞)(1, \infty) and is increasing for all x>1x > 1. However, it does not satisfy the third feature, as f(x)f(x) approaches positive infinity as xx approaches 1 from the right.

Function 2: f(x)=1x−2f(x) = \frac{1}{x-2}

This function has a domain of (2,∞)(2, \infty) and is increasing for all x>2x > 2. However, it does not satisfy the first feature, as f(x)f(x) is decreasing for all x<2x < 2.

Function 3: f(x)=1x−1+1f(x) = \frac{1}{x-1} + 1

This function has a domain of (1,∞)(1, \infty) and is increasing for all x>1x > 1. Additionally, it satisfies the third feature, as f(x)f(x) approaches negative infinity as xx approaches 1 from the right.

Function 4: f(x)=1x−2+1f(x) = \frac{1}{x-2} + 1

This function has a domain of (2,∞)(2, \infty) and is increasing for all x>2x > 2. However, it does not satisfy the first feature, as f(x)f(x) is decreasing for all x<2x < 2.

Function 5: f(x)=1x−1+1x−2f(x) = \frac{1}{x-1} + \frac{1}{x-2}

This function has a domain of (1,∞)(1, \infty) and is increasing for all x>1x > 1. Additionally, it satisfies the third feature, as f(x)f(x) approaches negative infinity as xx approaches 1 from the right.

Conclusion

Based on the analysis, we can conclude that the functions f(x)=1x−1+1f(x) = \frac{1}{x-1} + 1 and f(x)=1x−1+1x−2f(x) = \frac{1}{x-1} + \frac{1}{x-2} satisfy all the given features. These functions have an increasing nature, a domain of (1,∞)(1, \infty), and values that move toward negative for low values of xx. Therefore, the correct answer is Function 3: f(x)=1x−1+1f(x) = \frac{1}{x-1} + 1 and Function 5: f(x)=1x−1+1x−2f(x) = \frac{1}{x-1} + \frac{1}{x-2}.

Recommendations

  • When analyzing functions, it is essential to consider all the given features and determine which function satisfies all the conditions.
  • The domain of a function is a critical aspect to consider when analyzing its behavior.
  • The behavior of a function can be determined by analyzing its graph or by using mathematical techniques such as calculus.

Final Thoughts

Introduction

In our previous article, we discussed the characteristics of a function that has an increasing nature, a domain of (1,∞)(1, \infty), and values that move toward negative for low values of xx. We analyzed several functions and determined that the functions f(x)=1x−1+1f(x) = \frac{1}{x-1} + 1 and f(x)=1x−1+1x−2f(x) = \frac{1}{x-1} + \frac{1}{x-2} satisfy all the given features. In this article, we will answer some frequently asked questions related to selecting the correct function.

Q: What is the importance of analyzing the domain of a function?

A: The domain of a function is a critical aspect to consider when analyzing its behavior. It determines the set of all possible input values for which the function is defined. In this case, the domain is (1,∞)(1, \infty), which means that the function is defined for all real numbers greater than 1.

Q: How can I determine if a function is increasing or decreasing?

A: To determine if a function is increasing or decreasing, you can analyze its graph or use mathematical techniques such as calculus. If the function has a positive slope, it is increasing. If the function has a negative slope, it is decreasing.

Q: What is the significance of the vertical asymptote in this problem?

A: The vertical asymptote is a critical aspect of this problem. It represents the value of xx at which the function approaches positive or negative infinity. In this case, the vertical asymptote is at x=1x = 1, and the function approaches negative infinity as xx approaches 1 from the right.

Q: Can you provide more examples of functions that satisfy the given features?

A: Yes, here are a few more examples of functions that satisfy the given features:

  • f(x)=1x−1+2f(x) = \frac{1}{x-1} + 2
  • f(x)=1x−1+1x−2+1f(x) = \frac{1}{x-1} + \frac{1}{x-2} + 1
  • f(x)=1x−1+1x−2+1x−3f(x) = \frac{1}{x-1} + \frac{1}{x-2} + \frac{1}{x-3}

Q: How can I apply this knowledge to real-world problems?

A: This knowledge can be applied to real-world problems in various fields, such as physics, engineering, and economics. For example, in physics, you may need to analyze the behavior of a function that represents the motion of an object. In engineering, you may need to analyze the behavior of a function that represents the stress on a material. In economics, you may need to analyze the behavior of a function that represents the demand for a product.

Q: What are some common mistakes to avoid when analyzing functions?

A: Some common mistakes to avoid when analyzing functions include:

  • Not considering the domain of the function
  • Not analyzing the behavior of the function at its vertical asymptotes
  • Not using mathematical techniques such as calculus to analyze the function
  • Not considering the real-world implications of the function's behavior

Conclusion

In conclusion, selecting the correct function requires a thorough analysis of its features and behavior. By considering all the given features and determining which function satisfies all the conditions, we can arrive at the correct answer. We hope that this Q&A article has provided you with a better understanding of the concepts and techniques involved in selecting the correct function.