What Is The Domain Of The Step Function $f(x) = \lceil 2x \rceil - 1$?A. $\{x \mid X \geq -1\}$B. $\{x \mid X \geq 1\}$C. $\{x \mid X \text{ Is An Integer}\}$D. $\{x \mid X \text{ Is A Real Number}\}$

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Introduction

The step function, also known as the ceiling function, is a mathematical function that rounds a given real number up to the nearest integer. In this article, we will explore the domain of the step function $f(x) = \lceil 2x \rceil - 1$. The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $x$ for which $f(x)$ is defined.

Understanding the Step Function

The step function $f(x) = \lceil 2x \rceil - 1$ is a composition of two functions: the ceiling function $\lceil x \rceil$ and the linear function $2x - 1$. The ceiling function $\lceil x \rceil$ rounds a given real number up to the nearest integer. For example, $\lceil 2.3 \rceil = 3$ and $\lceil -2.3 \rceil = -2$. The linear function $2x - 1$ is a simple linear function that takes a real number $x$ and returns $2x - 1$.

Analyzing the Domain of the Step Function

To determine the domain of the step function $f(x) = \lceil 2x \rceil - 1$, we need to consider the possible values of $x$ for which the function is defined. Since the ceiling function $\lceil x \rceil$ is defined for all real numbers $x$, the only restriction on the domain of the step function comes from the linear function $2x - 1$. However, since the linear function $2x - 1$ is defined for all real numbers $x$, the domain of the step function $f(x) = \lceil 2x \rceil - 1$ is also the set of all real numbers.

Examining the Options

Let's examine the options given in the problem:

A. $\{x \mid x \geq -1\}$

This option suggests that the domain of the step function is the set of all real numbers greater than or equal to $-1$. However, as we have seen, the domain of the step function is actually the set of all real numbers.

B. $\{x \mid x \geq 1\}$

This option suggests that the domain of the step function is the set of all real numbers greater than or equal to $1$. However, as we have seen, the domain of the step function is actually the set of all real numbers.

C. $\{x \mid x \text{ is an integer}\}$

This option suggests that the domain of the step function is the set of all integers. However, as we have seen, the domain of the step function is actually the set of all real numbers.

D. $\{x \mid x \text{ is a real number}\}$

This option suggests that the domain of the step function is the set of all real numbers. This is the correct answer.

Conclusion

In conclusion, the domain of the step function $f(x) = \lceil 2x \rceil - 1$ is the set of all real numbers. This is because the ceiling function $\lceil x \rceil$ is defined for all real numbers $x$, and the linear function $2x - 1$ is also defined for all real numbers $x$. Therefore, the correct answer is option D: $\{x \mid x \text{ is a real number}\}$.

Final Answer

The final answer is $\boxed{D}$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Understand the step function $f(x) = \lceil 2x \rceil - 1$.
2. Analyze the domain of the step function.
3. Examine the options given in the problem.
4. Choose the correct answer.

Key Concepts

• The step function $f(x) = \lceil 2x \rceil - 1$ is a composition of the ceiling function $\lceil x \rceil$ and the linear function $2x - 1$.
• The ceiling function $\lceil x \rceil$ is defined for all real numbers $x$.
• The linear function $2x - 1$ is defined for all real numbers $x$.
• The domain of the step function $f(x) = \lceil 2x \rceil - 1$ is the set of all real numbers.

Common Mistakes

• Assuming that the domain of the step function is the set of all integers.
• Assuming that the domain of the step function is the set of all real numbers greater than or equal to $-1$.
• Assuming that the domain of the step function is the set of all real numbers greater than or equal to $1$.

Tips and Tricks

• Make sure to understand the step function $f(x) = \lceil 2x \rceil - 1$.
• Analyze the domain of the step function carefully.
• Examine the options given in the problem carefully.
• Choose the correct answer based on the analysis.
  

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Introduction

In our previous article, we explored the domain of the step function $f(x) = \lceil 2x \rceil - 1$. In this article, we will answer some frequently asked questions about the domain of the step function.

Q1: What is the domain of the step function $f(x) = \lceil 2x \rceil - 1$?

A1: The domain of the step function $f(x) = \lceil 2x \rceil - 1$ is the set of all real numbers.

Q2: Why is the domain of the step function $f(x) = \lceil 2x \rceil - 1$ the set of all real numbers?

A2: The domain of the step function $f(x) = \lceil 2x \rceil - 1$ is the set of all real numbers because the ceiling function $\lceil x \rceil$ is defined for all real numbers $x$, and the linear function $2x - 1$ is also defined for all real numbers $x$.

Q3: Is the domain of the step function $f(x) = \lceil 2x \rceil - 1$ the set of all integers?

A3: No, the domain of the step function $f(x) = \lceil 2x \rceil - 1$ is not the set of all integers. While the step function $f(x) = \lceil 2x \rceil - 1$ takes on integer values for integer inputs, it is defined for all real numbers.

Q4: Is the domain of the step function $f(x) = \lceil 2x \rceil - 1$ the set of all real numbers greater than or equal to $-1$?

A4: No, the domain of the step function $f(x) = \lceil 2x \rceil - 1$ is not the set of all real numbers greater than or equal to $-1$. The step function $f(x) = \lceil 2x \rceil - 1$ is defined for all real numbers, not just those greater than or equal to $-1$.

Q5: Is the domain of the step function $f(x) = \lceil 2x \rceil - 1$ the set of all real numbers greater than or equal to $1$?

A5: No, the domain of the step function $f(x) = \lceil 2x \rceil - 1$ is not the set of all real numbers greater than or equal to $1$. The step function $f(x) = \lceil 2x \rceil - 1$ is defined for all real numbers, not just those greater than or equal to $1$.

Q6: How do I determine the domain of a step function?

A6: To determine the domain of a step function, you need to analyze the function and determine the set of all possible input values for which the function is defined.

Q7: What are some common mistakes to avoid when determining the domain of a step function?

A7: Some common mistakes to avoid when determining the domain of a step function include assuming that the domain is the set of all integers, assuming that the domain is the set of all real numbers greater than or equal to $-1$, and assuming that the domain is the set of all real numbers greater than or equal to $1$.

Q8: What are some tips and tricks for determining the domain of a step function?

A8: Some tips and tricks for determining the domain of a step function include making sure to understand the function, analyzing the function carefully, and examining the options given in the problem carefully.

Conclusion

In conclusion, the domain of the step function $f(x) = \lceil 2x \rceil - 1$ is the set of all real numbers. We hope that this Q&A article has been helpful in answering some frequently asked questions about the domain of the step function.

Final Answer

The final answer is $\boxed{D}$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Understand the step function $f(x) = \lceil 2x \rceil - 1$.
2. Analyze the domain of the step function.
3. Examine the options given in the problem.
4. Choose the correct answer.

Key Concepts

• The step function $f(x) = \lceil 2x \rceil - 1$ is a composition of the ceiling function $\lceil x \rceil$ and the linear function $2x - 1$.
• The ceiling function $\lceil x \rceil$ is defined for all real numbers $x$.
• The linear function $2x - 1$ is defined for all real numbers $x$.
• The domain of the step function $f(x) = \lceil 2x \rceil - 1$ is the set of all real numbers.

Common Mistakes

• Assuming that the domain of the step function is the set of all integers.
• Assuming that the domain of the step function is the set of all real numbers greater than or equal to $-1$.
• Assuming that the domain of the step function is the set of all real numbers greater than or equal to $1$.

Tips and Tricks

• Make sure to understand the step function $f(x) = \lceil 2x \rceil - 1$.
• Analyze the domain of the step function carefully.
• Examine the options given in the problem carefully.
• Choose the correct answer based on the analysis.