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To determine which function has the same domain as [tex]\( y = 2 \sqrt{x} \)[/tex], we need to analyze the domain of [tex]\( y = 2 \sqrt{x} \)[/tex] and compare it to the domains of the other given functions.
### Step 1: Determine the domain of [tex]\( y = 2 \sqrt{x} \)[/tex]
The function [tex]\( y = 2 \sqrt{x} \)[/tex] contains a square root. A square root function is defined only when the expression inside the square root is non-negative. Thus:
[tex]\[ 2 \sqrt{x} \quad \text{is defined if} \quad x \geq 0 \][/tex]
So, the domain of [tex]\( y = 2 \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
### Step 2: Analyze the domains of the other functions
1. For [tex]\( y = \sqrt{2x} \)[/tex]:
[tex]\[ \sqrt{2x} \quad \text{is defined if} \quad 2x \geq 0 \][/tex]
Simplifying this inequality:
[tex]\[ x \geq 0 \][/tex]
So, the domain of [tex]\( y = \sqrt{2x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
2. For [tex]\( y = 2 \sqrt[3]{x} \)[/tex]:
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers because any real number can be cubed.
Therefore, the domain of [tex]\( y = 2 \sqrt[3]{x} \)[/tex] is all real numbers ([tex]\( -\infty < x < \infty \)[/tex]).
3. For [tex]\( y = \sqrt{x-2} \)[/tex]:
[tex]\[ \sqrt{x-2} \quad \text{is defined if} \quad x-2 \geq 0 \][/tex]
Simplifying this inequality:
[tex]\[ x \geq 2 \][/tex]
So, the domain of [tex]\( y = \sqrt{x-2} \)[/tex] is [tex]\( x \geq 2 \)[/tex].
4. For [tex]\( y = \sqrt[3]{x-2} \)[/tex]:
The cube root function [tex]\( \sqrt[3]{x-2} \)[/tex] is defined for all real numbers, regardless of the value of [tex]\( x-2 \)[/tex]. This is because any real number can be cubed.
Therefore, the domain of [tex]\( y = \sqrt[3]{x-2} \)[/tex] is all real numbers ([tex]\( -\infty < x < \infty \)[/tex]).
### Step 3: Compare domains
We compare each domain to the domain of [tex]\( y = 2 \sqrt{x} \)[/tex] (which is [tex]\( x \geq 0 \)[/tex]):
- [tex]\( y = \sqrt{2 x} \)[/tex] has [tex]\( x \geq 0 \)[/tex] — matches with [tex]\( y = 2 \sqrt{x} \)[/tex].
- [tex]\( y = 2 \sqrt[3]{x} \)[/tex] has all real numbers — does not match.
- [tex]\( y = \sqrt{x-2} \)[/tex] has [tex]\( x \geq 2 \)[/tex] — does not match.
- [tex]\( y = \sqrt[3]{x-2} \)[/tex] has all real numbers — does not match.
Thus, the function that has the same domain as [tex]\( y = 2 \sqrt{x} \)[/tex] is:
[tex]\[ y = \sqrt{2 x} \][/tex]
### Step 1: Determine the domain of [tex]\( y = 2 \sqrt{x} \)[/tex]
The function [tex]\( y = 2 \sqrt{x} \)[/tex] contains a square root. A square root function is defined only when the expression inside the square root is non-negative. Thus:
[tex]\[ 2 \sqrt{x} \quad \text{is defined if} \quad x \geq 0 \][/tex]
So, the domain of [tex]\( y = 2 \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
### Step 2: Analyze the domains of the other functions
1. For [tex]\( y = \sqrt{2x} \)[/tex]:
[tex]\[ \sqrt{2x} \quad \text{is defined if} \quad 2x \geq 0 \][/tex]
Simplifying this inequality:
[tex]\[ x \geq 0 \][/tex]
So, the domain of [tex]\( y = \sqrt{2x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
2. For [tex]\( y = 2 \sqrt[3]{x} \)[/tex]:
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers because any real number can be cubed.
Therefore, the domain of [tex]\( y = 2 \sqrt[3]{x} \)[/tex] is all real numbers ([tex]\( -\infty < x < \infty \)[/tex]).
3. For [tex]\( y = \sqrt{x-2} \)[/tex]:
[tex]\[ \sqrt{x-2} \quad \text{is defined if} \quad x-2 \geq 0 \][/tex]
Simplifying this inequality:
[tex]\[ x \geq 2 \][/tex]
So, the domain of [tex]\( y = \sqrt{x-2} \)[/tex] is [tex]\( x \geq 2 \)[/tex].
4. For [tex]\( y = \sqrt[3]{x-2} \)[/tex]:
The cube root function [tex]\( \sqrt[3]{x-2} \)[/tex] is defined for all real numbers, regardless of the value of [tex]\( x-2 \)[/tex]. This is because any real number can be cubed.
Therefore, the domain of [tex]\( y = \sqrt[3]{x-2} \)[/tex] is all real numbers ([tex]\( -\infty < x < \infty \)[/tex]).
### Step 3: Compare domains
We compare each domain to the domain of [tex]\( y = 2 \sqrt{x} \)[/tex] (which is [tex]\( x \geq 0 \)[/tex]):
- [tex]\( y = \sqrt{2 x} \)[/tex] has [tex]\( x \geq 0 \)[/tex] — matches with [tex]\( y = 2 \sqrt{x} \)[/tex].
- [tex]\( y = 2 \sqrt[3]{x} \)[/tex] has all real numbers — does not match.
- [tex]\( y = \sqrt{x-2} \)[/tex] has [tex]\( x \geq 2 \)[/tex] — does not match.
- [tex]\( y = \sqrt[3]{x-2} \)[/tex] has all real numbers — does not match.
Thus, the function that has the same domain as [tex]\( y = 2 \sqrt{x} \)[/tex] is:
[tex]\[ y = \sqrt{2 x} \][/tex]
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