Question

which function has the same domain as $y = 2\sqrt{x}$?
$y = \sqrt{x - 2}$ $y = \sqrt{2x}$ $y = \sqrt3{x - 2}$
$y = 2\sqrt3{x}$

Explanation:

Step1: Find domain of \( y = 2\sqrt{x} \)

For square root function, the expression inside (radicand) must be non - negative. So for \( y = 2\sqrt{x} \), we have \( x\geq0 \).

Step2: Find domain of \( y=\sqrt{x - 2} \)

The radicand \( x - 2\geq0\), so \( x\geq2 \).

Step3: Find domain of \( y=\sqrt{2x} \)

The radicand \( 2x\geq0\), dividing both sides by 2 (since 2>0, inequality direction remains same), we get \( x\geq0 \).

Step4: Find domain of \( y=\sqrt[3]{x - 2} \)

For cube root function, the radicand can be any real number (positive, negative, or zero). So the domain is all real numbers (\( x\in R \)).

Step5: Find domain of \( y = 2\sqrt[3]{x} \)

For cube root function, the radicand \( x\) can be any real number. So the domain is all real numbers (\( x\in R \)).

Answer:

\( y=\sqrt{2x} \) has the same domain as \( y = 2\sqrt{x} \)