Question

which graph represents ( y = sqrt3{x} )?

Explanation:

Step1: Analyze the function \( y = \sqrt[3]{x} \)

The cube - root function \( y=\sqrt[3]{x} \) has the following properties:

• Domain: All real numbers (\( x\in R \)), because we can take the cube - root of any real number (positive, negative, or zero).
• Range: All real numbers (\( y\in R \)), since for any real number \( y \), we can find an \( x=y^{3} \) such that \( \sqrt[3]{x}=y \).
• Symmetry: The function \( y = \sqrt[3]{x} \) is an odd function, which means \( f(-x)=-f(x) \). For example, when \( x = 8 \), \( y=\sqrt[3]{8} = 2 \); when \( x=-8 \), \( y=\sqrt[3]{-8}=-2 \). The graph of an odd function is symmetric about the origin.
• Key points: When \( x = 0 \), \( y=\sqrt[3]{0}=0 \); when \( x = 1 \), \( y=\sqrt[3]{1}=1 \); when \( x=-1 \), \( y=\sqrt[3]{-1}=-1 \); when \( x = 8 \), \( y = 2 \); when \( x=-8 \), \( y=-2 \).

Step2: Analyze the given graphs

• The first graph (the upper one) looks like a parabola \( y = x^{2} \) (it is symmetric about the \( y \) - axis, and for \( x\) positive and negative, \( y\) is non - negative). But the cube - root function is symmetric about the origin, not the \( y \) - axis, and takes negative values when \( x\) is negative. So this graph does not represent \( y=\sqrt[3]{x} \).
• The second graph (the lower one, although the description is a bit incomplete, we can infer from the properties of \( y = \sqrt[3]{x} \)): The cube - root function should pass through the origin, and for negative \( x \), \( y \) is negative, for positive \( x \), \( y \) is positive. If we assume the lower graph has the correct shape (passing through the origin, increasing for \( x>0 \), decreasing for \( x < 0 \), symmetric about the origin), it is more likely to represent \( y=\sqrt[3]{x} \). But since the user's graph description is a bit unclear, we can also think in terms of the general shape of \( y=\sqrt[3]{x} \): it has a point at the origin, and as \( x \) increases from \( -\infty \) to \( 0 \), \( y \) decreases from \( +\infty \) to \( 0 \), and as \( x \) increases from \( 0 \) to \( +\infty \), \( y \) increases from \( 0 \) to \( +\infty \). The upper graph is a parabola - like shape (even function), so it is not the cube - root function. If we have to choose between the given graphs (assuming the lower one is the one with the correct cubic - root shape), the correct graph for \( y=\sqrt[3]{x} \) is the lower graph (the one that is not the parabola - shaped one).

Answer:

The lower graph (the one that is not the parabola - shaped graph) represents \( y=\sqrt[3]{x} \)