A Step Function $h(x$\] Is Represented By $y = -2\lfloor X\rfloor$. Which Phrase Best Describes The Range Of The Function $h(x$\]?A. All Real Numbers B. All Rational Numbers C. All Negative Integers D. All Even Integers

A step function $h(x)$ is represented by the equation $y = -2\lfloor x\rfloor$. In this equation, $\lfloor x\rfloor$ represents the greatest integer less than or equal to $x$, also known as the floor function. This function is a type of step function, which is a piecewise function that has a finite number of discontinuities.

Analyzing the Function $h(x)$

To understand the range of the function $h(x)$, we need to analyze its behavior for different values of $x$. Since the function is defined as $y = -2\lfloor x\rfloor$, we can see that the output of the function will be an integer multiple of $-2$. This is because the floor function will always return an integer, and multiplying it by $-2$ will result in an integer multiple of $-2$.

Range of the Function $h(x)$

Now, let's consider the possible values of $y$ for different values of $x$. Since the floor function will always return an integer, we can see that the output of the function will be an integer multiple of $-2$. This means that the range of the function $h(x)$ will be all negative integers.

Why is the Range all Negative Integers?

To understand why the range of the function $h(x)$ is all negative integers, let's consider the behavior of the floor function. The floor function will always return the greatest integer less than or equal to $x$. This means that for any value of $x$, the output of the floor function will be an integer. Multiplying this integer by $-2$ will result in a negative integer.

Why is the Range not all Real Numbers?

One might argue that the range of the function $h(x)$ should be all real numbers, since the output of the function can be any real number. However, this is not the case. The output of the function is always an integer multiple of $-2$, which means that it can only take on integer values. Therefore, the range of the function $h(x)$ is not all real numbers.

Why is the Range not all Rational Numbers?

Another possibility is that the range of the function $h(x)$ is all rational numbers. However, this is not the case. The output of the function is always an integer multiple of $-2$, which means that it can only take on integer values. Rational numbers include fractions, which are not integers. Therefore, the range of the function $h(x)$ is not all rational numbers.

Why is the Range not all Even Integers?

Finally, one might argue that the range of the function $h(x)$ is all even integers. However, this is not the case. The output of the function is always an integer multiple of $-2$, which means that it can take on both positive and negative integer values. Therefore, the range of the function $h(x)$ is not all even integers.

Conclusion

In conclusion, the range of the function $h(x)$ is all negative integers. This is because the output of the function is always an integer multiple of $-2$, which means that it can only take on integer values. The range of the function is not all real numbers, all rational numbers, or all even integers.

Answer

The correct answer is C. all negative integers.

References

• [1] "Step Functions" by MathWorld
• [2] "Floor Function" by Wolfram MathWorld
• [3] "Range of a Function" by Math Open Reference
  Q&A: Understanding the Step Function $h(x)$ =====================================================

In the previous article, we discussed the step function $h(x)$ represented by the equation $y = -2\lfloor x\rfloor$. We analyzed the behavior of the function and concluded that the range of the function $h(x)$ is all negative integers. In this article, we will answer some frequently asked questions about the step function $h(x)$.

Q: What is the floor function?

A: The floor function, denoted by $\lfloor x\rfloor$, is a mathematical function that returns the greatest integer less than or equal to $x$. For example, $\lfloor 3.7\rfloor = 3$ and $\lfloor -2.3\rfloor = -3$.

Q: What is a step function?

A: A step function is a type of piecewise function that has a finite number of discontinuities. It is a function that has a finite number of distinct values, and the function changes value at a finite number of points.

Q: Why is the range of the function $h(x)$ all negative integers?

A: The range of the function $h(x)$ is all negative integers because the output of the function is always an integer multiple of $-2$. This means that the function can only take on integer values, and since the function is multiplied by $-2$, the values will always be negative.

Q: Can the function $h(x)$ take on any value in the range of all negative integers?

A: Yes, the function $h(x)$ can take on any value in the range of all negative integers. Since the function is defined as $y = -2\lfloor x\rfloor$, the output of the function will always be an integer multiple of $-2$, which means that it can take on any value in the range of all negative integers.

Q: Is the function $h(x)$ a continuous function?

A: No, the function $h(x)$ is not a continuous function. Since the function is a step function, it has a finite number of discontinuities, which means that it is not continuous.

Q: Can the function $h(x)$ be used to model real-world phenomena?

A: Yes, the function $h(x)$ can be used to model real-world phenomena. For example, the function can be used to model the behavior of a population that is decreasing at a constant rate.

Q: How can the function $h(x)$ be used in applications?

A: The function $h(x)$ can be used in a variety of applications, including modeling population growth and decline, modeling the behavior of physical systems, and modeling the behavior of economic systems.

Q: Can the function $h(x)$ be used to solve problems in mathematics?

A: Yes, the function $h(x)$ can be used to solve problems in mathematics. For example, the function can be used to solve problems involving the floor function, and it can be used to solve problems involving step functions.

Conclusion

In conclusion, the step function $h(x)$ is a type of piecewise function that has a finite number of discontinuities. The range of the function $h(x)$ is all negative integers, and the function can be used to model real-world phenomena and solve problems in mathematics.

References

• [1] "Step Functions" by MathWorld
• [2] "Floor Function" by Wolfram MathWorld
• [3] "Range of a Function" by Math Open Reference