Answer :
To determine which function has the same domain as [tex]\( y = 2 \sqrt{x} \)[/tex], we need to analyze the domains of each given function individually.
### Domain of [tex]\( y = 2 \sqrt{x} \)[/tex]
The function [tex]\( y = 2 \sqrt{x} \)[/tex] involves the square root of [tex]\( x \)[/tex]. The square root function [tex]\( \sqrt{x} \)[/tex] is defined for all [tex]\( x \)[/tex] that are greater than or equal to 0. Therefore, the domain of [tex]\( y = 2 \sqrt{x} \)[/tex] is:
[tex]\[ x \geq 0 \][/tex]
### Analyzing the domain of each given function:
1. [tex]\( y = \sqrt{2x} \)[/tex]
- This function involves the square root of [tex]\( 2x \)[/tex]. For the square root function [tex]\( \sqrt{2x} \)[/tex] to be defined, the argument [tex]\( 2x \)[/tex] must be non-negative, i.e., [tex]\( 2x \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[ 2x \geq 0 \implies x \geq 0 \][/tex]
- Domain of [tex]\( y = \sqrt{2x} \)[/tex]: [tex]\( x \geq 0 \)[/tex]
2. [tex]\( y = 2 \sqrt[3]{x} \)[/tex]
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- There are no restrictions on [tex]\( x \)[/tex] for [tex]\( y = 2 \sqrt[3]{x} \)[/tex].
- Domain of [tex]\( y = 2 \sqrt[3]{x} \)[/tex]: [tex]\( -\infty < x < \infty \)[/tex]
3. [tex]\( y = \sqrt{x - 2} \)[/tex]
- This function involves the square root of [tex]\( x - 2 \)[/tex]. For the square root function [tex]\( \sqrt{x - 2} \)[/tex] to be defined, the argument [tex]\( x - 2 \)[/tex] must be non-negative, i.e., [tex]\( x - 2 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[ x - 2 \geq 0 \implies x \geq 2 \][/tex]
- Domain of [tex]\( y = \sqrt{x - 2} \)[/tex]: [tex]\( x \geq 2 \)[/tex]
4. [tex]\( y = \sqrt[3]{x - 2} \)[/tex]
- The cube root function [tex]\( \sqrt[3]{x - 2} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- There are no restrictions on [tex]\( x \)[/tex] for [tex]\( y = \sqrt[3]{x - 2} \)[/tex].
- Domain of [tex]\( y = \sqrt[3]{x - 2} \)[/tex]: [tex]\( -\infty < x < \infty \)[/tex]
### Conclusion:
The function [tex]\( y = 2 \sqrt{x} \)[/tex] has the domain [tex]\( x \geq 0 \)[/tex]. Among the given functions, the function that shares this domain is:
[tex]\[ y = \sqrt{2x} \][/tex]
Therefore, the function with the same domain as [tex]\( y = 2 \sqrt{x} \)[/tex] is:
[tex]\[ \boxed{y = \sqrt{2x}} \][/tex]
### Domain of [tex]\( y = 2 \sqrt{x} \)[/tex]
The function [tex]\( y = 2 \sqrt{x} \)[/tex] involves the square root of [tex]\( x \)[/tex]. The square root function [tex]\( \sqrt{x} \)[/tex] is defined for all [tex]\( x \)[/tex] that are greater than or equal to 0. Therefore, the domain of [tex]\( y = 2 \sqrt{x} \)[/tex] is:
[tex]\[ x \geq 0 \][/tex]
### Analyzing the domain of each given function:
1. [tex]\( y = \sqrt{2x} \)[/tex]
- This function involves the square root of [tex]\( 2x \)[/tex]. For the square root function [tex]\( \sqrt{2x} \)[/tex] to be defined, the argument [tex]\( 2x \)[/tex] must be non-negative, i.e., [tex]\( 2x \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[ 2x \geq 0 \implies x \geq 0 \][/tex]
- Domain of [tex]\( y = \sqrt{2x} \)[/tex]: [tex]\( x \geq 0 \)[/tex]
2. [tex]\( y = 2 \sqrt[3]{x} \)[/tex]
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- There are no restrictions on [tex]\( x \)[/tex] for [tex]\( y = 2 \sqrt[3]{x} \)[/tex].
- Domain of [tex]\( y = 2 \sqrt[3]{x} \)[/tex]: [tex]\( -\infty < x < \infty \)[/tex]
3. [tex]\( y = \sqrt{x - 2} \)[/tex]
- This function involves the square root of [tex]\( x - 2 \)[/tex]. For the square root function [tex]\( \sqrt{x - 2} \)[/tex] to be defined, the argument [tex]\( x - 2 \)[/tex] must be non-negative, i.e., [tex]\( x - 2 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[ x - 2 \geq 0 \implies x \geq 2 \][/tex]
- Domain of [tex]\( y = \sqrt{x - 2} \)[/tex]: [tex]\( x \geq 2 \)[/tex]
4. [tex]\( y = \sqrt[3]{x - 2} \)[/tex]
- The cube root function [tex]\( \sqrt[3]{x - 2} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
- There are no restrictions on [tex]\( x \)[/tex] for [tex]\( y = \sqrt[3]{x - 2} \)[/tex].
- Domain of [tex]\( y = \sqrt[3]{x - 2} \)[/tex]: [tex]\( -\infty < x < \infty \)[/tex]
### Conclusion:
The function [tex]\( y = 2 \sqrt{x} \)[/tex] has the domain [tex]\( x \geq 0 \)[/tex]. Among the given functions, the function that shares this domain is:
[tex]\[ y = \sqrt{2x} \][/tex]
Therefore, the function with the same domain as [tex]\( y = 2 \sqrt{x} \)[/tex] is:
[tex]\[ \boxed{y = \sqrt{2x}} \][/tex]
