3. Consider The Function $f(x) = \frac{1}{x-2} + 1$.- Vertical Asymptote (VA): $x = 2$- Horizontal Asymptote (HA): $y = 1$Domain: $(-\infty, 2) \cup (2, \infty$\]Range: $(-\infty, 1) \cup (1, \infty$\]

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Introduction

In mathematics, functions are used to describe the relationship between variables. A function can be represented in various forms, including algebraic, trigonometric, and exponential forms. In this article, we will focus on the function $f(x) = \frac{1}{x-2} + 1$ and explore its properties, including vertical and horizontal asymptotes, domain, and range.

Vertical Asymptote (VA)

A vertical asymptote is a vertical line that a function approaches but never touches. In the case of the function $f(x) = \frac{1}{x-2} + 1$, the vertical asymptote is $x = 2$. This is because when $x$ approaches 2 from the left or right, the denominator of the fraction approaches 0, causing the function to approach infinity.

The concept of a vertical asymptote is crucial in understanding the behavior of a function.

Horizontal Asymptote (HA)

A horizontal asymptote is a horizontal line that a function approaches as $x$ approaches infinity or negative infinity. In the case of the function $f(x) = \frac{1}{x-2} + 1$, the horizontal asymptote is $y = 1$. This is because as $x$ approaches infinity or negative infinity, the fraction $\frac{1}{x-2}$ approaches 0, causing the function to approach 1.

The concept of a horizontal asymptote is essential in understanding the behavior of a function as it approaches infinity or negative infinity.

Domain

The domain of a function is the set of all possible input values for which the function is defined. In the case of the function $f(x) = \frac{1}{x-2} + 1$, the domain is $(-\infty, 2) \cup (2, \infty)$. This means that the function is defined for all real numbers except $x = 2$, where the denominator of the fraction is 0.

The concept of a domain is crucial in understanding the behavior of a function.

Range

The range of a function is the set of all possible output values for which the function is defined. In the case of the function $f(x) = \frac{1}{x-2} + 1$, the range is $(-\infty, 1) \cup (1, \infty)$. This means that the function can take on any real value except $y = 1$, where the function approaches a horizontal asymptote.

The concept of a range is essential in understanding the behavior of a function.

Graphing the Function

To visualize the behavior of the function $f(x) = \frac{1}{x-2} + 1$, we can graph the function. The graph will show the vertical asymptote at $x = 2$, the horizontal asymptote at $y = 1$, and the domain and range of the function.

Graphing the function is a useful tool in understanding its behavior.

Conclusion

In conclusion, the function $f(x) = \frac{1}{x-2} + 1$ has a vertical asymptote at $x = 2$, a horizontal asymptote at $y = 1$, a domain of $(-\infty, 2) \cup (2, \infty)$, and a range of $(-\infty, 1) \cup (1, \infty)$. Understanding these properties is essential in understanding the behavior of the function.

Understanding the properties of a function is crucial in mathematics.

References

• [1] Calculus, 3rd edition, by Michael Spivak
• [2] Algebra, 2nd edition, by Michael Artin
• [3] Graphing Functions, by Michael Sullivan

Further Reading

• [1] Asymptotes, by Math Open Reference
• [2] Domain and Range, by Math Is Fun
• [3] Graphing Functions, by Khan Academy
  

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Q: What is the vertical asymptote of the function $f(x) = \frac{1}{x-2} + 1$?

A: The vertical asymptote of the function $f(x) = \frac{1}{x-2} + 1$ is $x = 2$. This is because when $x$ approaches 2 from the left or right, the denominator of the fraction approaches 0, causing the function to approach infinity.

Q: What is the horizontal asymptote of the function $f(x) = \frac{1}{x-2} + 1$?

A: The horizontal asymptote of the function $f(x) = \frac{1}{x-2} + 1$ is $y = 1$. This is because as $x$ approaches infinity or negative infinity, the fraction $\frac{1}{x-2}$ approaches 0, causing the function to approach 1.

Q: What is the domain of the function $f(x) = \frac{1}{x-2} + 1$?

A: The domain of the function $f(x) = \frac{1}{x-2} + 1$ is $(-\infty, 2) \cup (2, \infty)$. This means that the function is defined for all real numbers except $x = 2$, where the denominator of the fraction is 0.

Q: What is the range of the function $f(x) = \frac{1}{x-2} + 1$?

A: The range of the function $f(x) = \frac{1}{x-2} + 1$ is $(-\infty, 1) \cup (1, \infty)$. This means that the function can take on any real value except $y = 1$, where the function approaches a horizontal asymptote.

Q: How do I graph the function $f(x) = \frac{1}{x-2} + 1$?

A: To graph the function $f(x) = \frac{1}{x-2} + 1$, you can use a graphing calculator or a computer program. You can also use a piece of graph paper and a pencil to draw the graph by hand. The graph will show the vertical asymptote at $x = 2$, the horizontal asymptote at $y = 1$, and the domain and range of the function.

Q: What is the significance of the vertical and horizontal asymptotes in the function $f(x) = \frac{1}{x-2} + 1$?

A: The vertical and horizontal asymptotes in the function $f(x) = \frac{1}{x-2} + 1$ are important because they help to understand the behavior of the function. The vertical asymptote at $x = 2$ indicates that the function approaches infinity as $x$ approaches 2 from the left or right. The horizontal asymptote at $y = 1$ indicates that the function approaches 1 as $x$ approaches infinity or negative infinity.

Q: Can I use the function $f(x) = \frac{1}{x-2} + 1$ in real-world applications?

A: Yes, the function $f(x) = \frac{1}{x-2} + 1$ can be used in real-world applications, such as modeling the behavior of a physical system or predicting the outcome of a mathematical model. However, the function is not defined at $x = 2$, so you would need to use a different function or a different approach to model the behavior of the system at that point.

Q: How do I find the inverse of the function $f(x) = \frac{1}{x-2} + 1$?

A: To find the inverse of the function $f(x) = \frac{1}{x-2} + 1$, you can use the following steps:

1. Replace $f(x)$ with $y$.
2. Switch $x$ and $y$.
3. Solve for $y$.

The inverse of the function $f(x) = \frac{1}{x-2} + 1$ is $f^{-1}(x) = \frac{1}{x-1} + 2$.

Q: Can I use the function $f(x) = \frac{1}{x-2} + 1$ in calculus?

A: Yes, the function $f(x) = \frac{1}{x-2} + 1$ can be used in calculus, such as finding the derivative or integral of the function. However, you would need to use the chain rule or other calculus techniques to find the derivative or integral of the function.

Q: How do I find the derivative of the function $f(x) = \frac{1}{x-2} + 1$?

A: To find the derivative of the function $f(x) = \frac{1}{x-2} + 1$, you can use the following steps:

1. Use the chain rule to find the derivative of the fraction.
2. Simplify the derivative.

The derivative of the function $f(x) = \frac{1}{x-2} + 1$ is $f'(x) = -\frac{1}{(x-2)^2}$.

Q: Can I use the function $f(x) = \frac{1}{x-2} + 1$ in statistics?

A: Yes, the function $f(x) = \frac{1}{x-2} + 1$ can be used in statistics, such as modeling the behavior of a probability distribution or predicting the outcome of a statistical model. However, you would need to use a different function or a different approach to model the behavior of the distribution or predict the outcome of the model.

Q: How do I find the expected value of the function $f(x) = \frac{1}{x-2} + 1$?

A: To find the expected value of the function $f(x) = \frac{1}{x-2} + 1$, you can use the following steps:

1. Find the probability density function (pdf) of the function.
2. Integrate the pdf with respect to $x$.

The expected value of the function $f(x) = \frac{1}{x-2} + 1$ is $E(X) = \int_{-\infty}^{\infty} x \cdot f(x) dx$.

Q: Can I use the function $f(x) = \frac{1}{x-2} + 1$ in engineering?

A: Yes, the function $f(x) = \frac{1}{x-2} + 1$ can be used in engineering, such as modeling the behavior of a physical system or predicting the outcome of a mathematical model. However, you would need to use a different function or a different approach to model the behavior of the system or predict the outcome of the model.

Q: How do I find the frequency response of the function $f(x) = \frac{1}{x-2} + 1$?

A: To find the frequency response of the function $f(x) = \frac{1}{x-2} + 1$, you can use the following steps:

1. Find the transfer function of the system.
2. Use the transfer function to find the frequency response.

The frequency response of the function $f(x) = \frac{1}{x-2} + 1$ is $H(j\omega) = \frac{1}{j\omega-2} + 1$.

Q: Can I use the function $f(x) = \frac{1}{x-2} + 1$ in computer science?

A: Yes, the function $f(x) = \frac{1}{x-2} + 1$ can be used in computer science, such as modeling the behavior of a computer system or predicting the outcome of a mathematical model. However, you would need to use a different function or a different approach to model the behavior of the system or predict the outcome of the model.

Q: How do I find the time complexity of the function $f(x) = \frac{1}{x-2} + 1$?

A: To find the time complexity of the function $f(x) = \frac{1}{x-2} + 1$, you can use the following steps:

1. Analyze the function to determine the number of operations required to evaluate it.
2. Use the number of operations to determine the time complexity.

The time complexity of the function $f(x) = \frac{1}{x-2} + 1$ is $O(1)$.

Q: Can I use the function $f(x) = \frac{1}{x-2} + 1$ in data analysis?

A: Yes, the function $f(x) = \frac{1}{x-2} + 1$ can be used in data analysis, such as modeling the behavior of a dataset or predicting the outcome of a mathematical model. However, you would need to use a different function or a different approach to model the behavior of the dataset or predict the outcome of the model.

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