Which Function Has The Same Domain As $y=2 \sqrt{x}$?A. $y=\sqrt{2x}$B. $y=2 \sqrt[3]{x}$C. $y=\sqrt{x-2}$D. $y=\sqrt[3]{x-2}$

Which Function Has the Same Domain as $y=2 \sqrt{x}$?

Understanding the Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. When dealing with functions involving square roots, the domain is restricted to non-negative values, as the square root of a negative number is undefined in the real number system.

Analyzing the Given Function

The given function is $y=2 \sqrt{x}$. To find the domain of this function, we need to consider the values of x for which the expression under the square root is non-negative. Since the square root of a negative number is undefined, we must have $x \geq 0$. This means that the domain of the function $y=2 \sqrt{x}$ is all non-negative real numbers, or $x \geq 0$.

Evaluating the Options

Now, let's evaluate the options to determine which function has the same domain as $y=2 \sqrt{x}$.

A. $y=\sqrt{2x}$

The domain of this function is restricted to $x \geq 0$, as the expression under the square root must be non-negative. However, the function $y=\sqrt{2x}$ has a different form than $y=2 \sqrt{x}$. To compare the domains, we can rewrite the function $y=2 \sqrt{x}$ as $y=2 \sqrt{x} = \sqrt{2x}$. This shows that the function $y=\sqrt{2x}$ has the same form as $y=2 \sqrt{x}$, but with a different coefficient. Since the domain of $y=\sqrt{2x}$ is also $x \geq 0$, this option is a strong candidate.

B. $y=2 \sqrt[3]{x}$

The domain of this function is all real numbers, as the cube root of any real number is defined. This means that the function $y=2 \sqrt[3]{x}$ has a different domain than $y=2 \sqrt{x}$, which is restricted to non-negative real numbers. Therefore, this option is not a good candidate.

C. $y=\sqrt{x-2}$

The domain of this function is restricted to $x \geq 2$, as the expression under the square root must be non-negative. This means that the function $y=\sqrt{x-2}$ has a different domain than $y=2 \sqrt{x}$, which is restricted to non-negative real numbers. Therefore, this option is not a good candidate.

D. $y=\sqrt[3]{x-2}$

The domain of this function is all real numbers, as the cube root of any real number is defined. This means that the function $y=\sqrt[3]{x-2}$ has a different domain than $y=2 \sqrt{x}$, which is restricted to non-negative real numbers. Therefore, this option is not a good candidate.

Conclusion

Based on the analysis, the function $y=\sqrt{2x}$ has the same domain as $y=2 \sqrt{x}$, which is all non-negative real numbers, or $x \geq 0$. This is because the function $y=\sqrt{2x}$ has the same form as $y=2 \sqrt{x}$, but with a different coefficient. Therefore, the correct answer is:

A. $y=\sqrt{2x}$

Understanding the Importance of Domain

The domain of a function is a critical aspect of understanding the behavior and properties of the function. It determines the set of input values for which the function is defined, and it can affect the function's graph, behavior, and applications. In this case, the domain of the function $y=2 \sqrt{x}$ is restricted to non-negative real numbers, which means that the function is only defined for values of x greater than or equal to 0.

Real-World Applications

The concept of domain is essential in various real-world applications, such as:

• Physics and Engineering: When modeling physical systems, the domain of a function can represent the set of possible input values, such as time, distance, or temperature.
• Economics: In economic models, the domain of a function can represent the set of possible input values, such as prices, quantities, or rates of change.
• Computer Science: In computer programming, the domain of a function can represent the set of possible input values, such as user input, data, or parameters.

Conclusion

In conclusion, the function $y=\sqrt{2x}$ has the same domain as $y=2 \sqrt{x}$, which is all non-negative real numbers, or $x \geq 0$. This is because the function $y=\sqrt{2x}$ has the same form as $y=2 \sqrt{x}$, but with a different coefficient. Understanding the domain of a function is essential in various real-world applications, and it can affect the function's graph, behavior, and applications.
Q&A: Understanding the Domain of a Function

Frequently Asked Questions

In this article, we will address some of the most common questions related to the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the values of x for which the expression under the square root is non-negative. If the expression under the square root is negative, the function is undefined for that value of x.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can a function have an empty domain?

A: Yes, a function can have an empty domain. This occurs when the expression under the square root is always negative, or when the function is undefined for all possible values of x.

Q: Can a function have a domain that is all real numbers?

A: Yes, a function can have a domain that is all real numbers. This occurs when the expression under the square root is always non-negative, or when the function is defined for all possible values of x.

Q: How does the domain of a function affect its graph?

A: The domain of a function affects its graph by determining the set of x-values that are included in the graph. If the domain of a function is restricted, the graph will only include those x-values that are within the domain.

Q: Can the domain of a function be changed?

A: Yes, the domain of a function can be changed by restricting or extending the set of x-values that are included in the domain. This can be done by adding or removing restrictions on the expression under the square root.

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

A: Understanding the domain of a function is essential in various real-world applications, such as physics, engineering, economics, and computer science. It can affect the function's graph, behavior, and applications.

Q: Can you provide examples of functions with different domains?

A: Yes, here are some examples of functions with different domains:

• Domain: all real numbers: $y = x^2$
• Domain: non-negative real numbers: $y = \sqrt{x}$
• Domain: negative real numbers: $y = -\sqrt{x}$
• Domain: empty set: $y = \sqrt{-x}$

Conclusion

In conclusion, understanding the domain of a function is essential in various real-world applications. It can affect the function's graph, behavior, and applications. By answering these frequently asked questions, we hope to have provided a better understanding of the domain of a function and its importance.

Additional Resources

For more information on the domain of a function, we recommend the following resources:

• Textbooks: "Calculus" by Michael Spivak, "Algebra" by Michael Artin
• Online resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Software: Mathematica, Maple, MATLAB

Final Thoughts

Understanding the domain of a function is a critical aspect of mathematics and its applications. By mastering this concept, you will be able to analyze and solve problems in various fields, from physics and engineering to economics and computer science.