Question

Compare the square root and cube root functions by completing the following:

Complete an input-output table for the functions f(x)=x−−√
and g(x)=x−−√3
. You do not need to submit the tables.
Graph the functions by plotting the points from the input-output table. You do not need to submit the graphs.
In 3–5 sentences, discuss the similarities and differences between the graphs of the functions.

Answer 1

Both f(x)=√x and g(x)=∛x are increasing root functions and meet at (0,0) and (1,1), but f has domain and range [0,∞) while g is defined for all real x with range ℝ. g(x) extends into negative x and is odd (symmetric about the origin); √x has no negative x-values. For 0<x<1 the cube root produces larger outputs (∛x > √x), but for x>1 the square root grows faster (√x > ∛x). Both graphs have very steep (vertical-tangent–like) behavior near 0 and are concave down for x>0, yet ∛x changes concavity at 0 (an inflection point) while √x remains concave down on its whole domain.