Question

which is the graph of the cube root function $f(x) = \sqrt3{x}$?

Explanation:

Step1: Recall cube root function properties

The cube root function \( f(x) = \sqrt[3]{x} \) is an odd function, so it is symmetric about the origin. Also, when \( x = 0 \), \( f(0) = 0 \); when \( x = 1 \), \( f(1) = 1 \); when \( x = -1 \), \( f(-1) = -1 \); as \( x \to +\infty \), \( f(x) \to +\infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

Step2: Analyze the two graphs

• First graph: For negative \( x \), it's in the upper half (positive \( y \)), for positive \( x \), it's in the lower half (negative \( y \)) – not symmetric about origin, and doesn't satisfy \( f(1)=1 \), \( f(-1)=-1 \).
• Second graph: Passes through \((0,0)\), \((1,1)\), \((-1,-1)\), and is symmetric about the origin (since odd function), matching the properties of \( f(x)=\sqrt[3]{x} \).

Answer:

The Middle Graph (the second graph shown, with the curve passing through (0,0), (1,1), (-1,-1) and symmetric about the origin)