Question

which of the following graphs are identical? choose two correct answers.
$y = -sqrt3{x}$
$y = sqrt{x}$
$y = sqrt3{x}$
$y = sqrt3{-x}$
$y = -sqrt{x}$
$y = sqrt{-x}$

Explanation:

Step1: Recall Cube Root Property

The cube root of a negative number can be rewritten: $\sqrt[3]{-x} = -\sqrt[3]{x}$ (since $\sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b}$ and $\sqrt[3]{-1}=-1$). So, $y = \sqrt[3]{-x}$ is equivalent to $y = -\sqrt[3]{x}$.

Step2: Compare Functions

Check the first function $y = -\sqrt[3]{x}$ and the fourth function $y=\sqrt[3]{-x}$. From the cube root property, they are identical. Other functions: $y = \sqrt{x}$ and $y = -\sqrt{x}$ are negatives, $y=\sqrt[3]{x}$ is positive cube root, $y=\sqrt{-x}$ has domain $x\leq0$ while $y=\sqrt{x}$ has $x\geq0$, so only $y = -\sqrt[3]{x}$ and $y=\sqrt[3]{-x}$ are identical.

Answer:

The graphs of $y = -\sqrt[3]{x}$ (top - left) and $y=\sqrt[3]{-x}$ (bottom - left) are identical.