Find The Range Of The Function:$\[ H: X \rightarrow \sqrt{2x - 8} \\]

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Introduction

In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. In this article, we will focus on finding the range of the function $h: x \rightarrow \sqrt{2x - 8}$. This function involves a square root, which means we need to consider the domain of the function and the properties of the square root function to determine its range.

Understanding the Domain of the Function

Before we can find the range of the function, we need to understand its domain. The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is defined as $h: x \rightarrow \sqrt{2x - 8}$. The expression inside the square root must be non-negative, so we have the inequality $2x - 8 \geq 0$. Solving this inequality, we get $x \geq 4$. Therefore, the domain of the function is $x \geq 4$.

Properties of the Square Root Function

The square root function has some important properties that we need to consider when finding the range of the function. One of the key properties is that the square root function is increasing, meaning that as the input value increases, the output value also increases. Another important property is that the square root function is always non-negative, meaning that the output value is always greater than or equal to zero.

Finding the Range of the Function

Now that we have understood the domain of the function and the properties of the square root function, we can find the range of the function. Since the function is defined as $h: x \rightarrow \sqrt{2x - 8}$, we can see that the output value is always non-negative. As the input value increases, the output value also increases. Therefore, the range of the function is all non-negative real numbers.

Mathematical Representation of the Range

Mathematically, we can represent the range of the function as $[0, \infty)$. This means that the range of the function includes all non-negative real numbers, from 0 to infinity.

Conclusion

In conclusion, we have found the range of the function $h: x \rightarrow \sqrt{2x - 8}$. The range of the function is all non-negative real numbers, which can be represented mathematically as $[0, \infty)$. This result is based on the properties of the square root function and the domain of the function.

Example Use Cases

The range of the function has several important implications in various fields, including mathematics, physics, and engineering. For example, in physics, the range of the function can be used to model the motion of objects under the influence of gravity. In engineering, the range of the function can be used to design systems that involve square root functions.

Final Thoughts

In this article, we have found the range of the function $h: x \rightarrow \sqrt{2x - 8}$. The range of the function is all non-negative real numbers, which can be represented mathematically as $[0, \infty)$. This result is based on the properties of the square root function and the domain of the function. We hope that this article has provided a clear understanding of the range of the function and its implications in various fields.

References

• [1] "Calculus" by Michael Spivak
• [2] "Mathematics for Engineers" by John R. Taylor
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler

Further Reading

For further reading on the topic of finding the range of functions, we recommend the following resources:

• [1] "Introduction to Calculus" by Gilbert Strang
• [2] "Mathematics for Computer Science" by Eric Lehman
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler

FAQs

• Q: What is the range of the function $h: x \rightarrow \sqrt{2x - 8}$? A: The range of the function is all non-negative real numbers, which can be represented mathematically as $[0, \infty)$.
• Q: What is the domain of the function $h: x \rightarrow \sqrt{2x - 8}$? A: The domain of the function is $x \geq 4$.
• Q: What are the properties of the square root function? A: The square root function is increasing and always non-negative.
  

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Introduction

In our previous article, we discussed how to find the range of the function $h: x \rightarrow \sqrt{2x - 8}$. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have. We will cover a range of topics, from the domain of the function to the properties of the square root function.

Q&A

Q: What is the domain of the function $h: x \rightarrow \sqrt{2x - 8}$?

A: The domain of the function is $x \geq 4$. This is because the expression inside the square root must be non-negative, so we have the inequality $2x - 8 \geq 0$. Solving this inequality, we get $x \geq 4$.

Q: What are the properties of the square root function?

A: The square root function is increasing and always non-negative. This means that as the input value increases, the output value also increases, and the output value is always greater than or equal to zero.

Q: How do I find the range of the function $h: x \rightarrow \sqrt{2x - 8}$?

A: To find the range of the function, we need to consider the domain of the function and the properties of the square root function. Since the function is defined as $h: x \rightarrow \sqrt{2x - 8}$, we can see that the output value is always non-negative. As the input value increases, the output value also increases. Therefore, the range of the function is all non-negative real numbers.

Q: What is the mathematical representation of the range of the function $h: x \rightarrow \sqrt{2x - 8}$?

A: The range of the function can be represented mathematically as $[0, \infty)$. This means that the range of the function includes all non-negative real numbers, from 0 to infinity.

Q: Can I use the range of the function $h: x \rightarrow \sqrt{2x - 8}$ in real-world applications?

A: Yes, the range of the function can be used in various real-world applications, such as modeling the motion of objects under the influence of gravity or designing systems that involve square root functions.

Q: How do I determine the domain of the function $h: x \rightarrow \sqrt{2x - 8}$?

A: To determine the domain of the function, we need to consider the expression inside the square root. The expression must be non-negative, so we have the inequality $2x - 8 \geq 0$. Solving this inequality, we get $x \geq 4$.

Q: What are some common mistakes to avoid when finding the range of a function?

A: Some common mistakes to avoid when finding the range of a function include:

• Not considering the domain of the function
• Not considering the properties of the square root function
• Not representing the range of the function mathematically

Conclusion

In this Q&A article, we have covered a range of topics related to finding the range of the function $h: x \rightarrow \sqrt{2x - 8}$. We hope that this article has provided a clear understanding of the domain of the function, the properties of the square root function, and the range of the function. If you have any further questions or doubts, please feel free to ask.

References

• [1] "Calculus" by Michael Spivak
• [2] "Mathematics for Engineers" by John R. Taylor
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler

Further Reading

For further reading on the topic of finding the range of functions, we recommend the following resources:

• [1] "Introduction to Calculus" by Gilbert Strang
• [2] "Mathematics for Computer Science" by Eric Lehman
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler

FAQs

• Q: What is the range of the function $h: x \rightarrow \sqrt{2x - 8}$? A: The range of the function is all non-negative real numbers, which can be represented mathematically as $[0, \infty)$.
• Q: What is the domain of the function $h: x \rightarrow \sqrt{2x - 8}$? A: The domain of the function is $x \geq 4$.
• Q: What are the properties of the square root function? A: The square root function is increasing and always non-negative.