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

Introduction

When dealing with functions, understanding their domains is crucial for accurate calculations and interpretations. The domain of a function represents the set of all possible input values (x-values) for which the function is defined. In this article, we will explore the domain of the function $y=2 \sqrt{x}$ and compare it with the given options to determine which function has the same domain.

Understanding the Domain of $y=2 \sqrt{x}$

The function $y=2 \sqrt{x}$ involves the square root of $x$. The square root of a number is defined only for non-negative real numbers. Therefore, the domain of $y=2 \sqrt{x}$ consists of all non-negative real numbers, which can be represented as $x \geq 0$.

Analyzing the Options

Now, let's analyze each of the given options to determine which one has the same domain as $y=2 \sqrt{x}$.

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

The function $y=\sqrt{2 x}$ involves the square root of $2x$. Since $2x$ is always non-negative for $x \geq 0$, the domain of this function is also $x \geq 0$. However, we need to consider the coefficient of $x$ inside the square root. If $x$ is negative, $2x$ will be negative, and the square root will be undefined. Therefore, the domain of this function is also $x \geq 0$.

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

The function $y=2 \sqrt[3]{x}$ involves the cube root of $x$. The cube root of a number is defined for all real numbers, regardless of their sign. Therefore, the domain of this function is all real numbers, which can be represented as $x \in \mathbb{R}$. This is different from the domain of $y=2 \sqrt{x}$, which is $x \geq 0$.

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

The function $y=\sqrt{x-2}$ involves the square root of $x-2$. For the square root to be defined, $x-2$ must be non-negative. Therefore, the domain of this function is $x \geq 2$. This is different from the domain of $y=2 \sqrt{x}$, which is $x \geq 0$.

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

The function $y=\sqrt[3]{x-2}$ involves the cube root of $x-2$. The cube root of a number is defined for all real numbers, regardless of their sign. Therefore, the domain of this function is all real numbers, which can be represented as $x \in \mathbb{R}$. This is different from the domain of $y=2 \sqrt{x}$, which is $x \geq 0$.

Conclusion

Based on the analysis of each option, we can conclude that only option A, $y=\sqrt{2 x}$, has the same domain as $y=2 \sqrt{x}$. Both functions have a domain of $x \geq 0$, which represents all non-negative real numbers.

Final Answer

Introduction

In our previous article, we explored the domain of the function $y=2 \sqrt{x}$ and compared it with the given options to determine which function has the same domain. In this article, we will answer some frequently asked questions related to the domain of functions.

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 values of x that can be plugged into the function.

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

A: To determine the domain of a function, you need to consider the restrictions on the input values. For example, if the function involves a square root, you need to ensure that the input value is non-negative. If the function involves a fraction, you need to ensure that the denominator is not equal to zero.

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, while the range of a function is the set of all possible output values. In other words, the domain is the set of all possible x-values, while the range is the set of all possible y-values.

Q: Can a function have a domain of all real numbers?

A: Yes, a function can have a domain of all real numbers. For example, the function $y=x^2$ has a domain of all real numbers, since any real number can be squared.

Q: Can a function have a domain of a single value?

A: Yes, a function can have a domain of a single value. For example, the function $y=\sqrt{x-4}$ has a domain of $x=4$, since the square root is only defined when $x-4$ is equal to zero.

Q: How do I graph a function with a restricted domain?

A: To graph a function with a restricted domain, you need to identify the restrictions on the input values and then graph the function accordingly. For example, if the function involves a square root, you need to ensure that the input value is non-negative.

Q: Can a function have a domain of all integers?

A: Yes, a function can have a domain of all integers. For example, the function $y=x^2$ has a domain of all integers, since any integer can be squared.

Q: Can a function have a domain of a set of discrete values?

A: Yes, a function can have a domain of a set of discrete values. For example, the function $y=\sqrt{x}$ has a domain of all non-negative integers, since the square root is only defined when the input value is non-negative.

Conclusion

In this article, we answered some frequently asked questions related to the domain of functions. We hope that this article has provided you with a better understanding of the domain of functions and how to determine it.

Final Answer

The final answer is that the domain of a function is the set of all possible input values (x-values) for which the function is defined.