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