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

Question

A) Complete an input-output table for the functions f(x)=x−−√
and g(x)=x−−√3
. You do not need to submit the tables.

B) Graph the functions by plotting the points from the input-output table. You do not need to submit the graphs.

C) In 3–5 sentences, discuss the similarities and differences between the graphs of the functions.

Answer 1

The square root function, \( f(x) = \sqrt{x} \), and the cube root function, \( g(x) = \sqrt[3]{x} \), exhibit distinct characteristics in their graphs.

Similarities: Both functions pass through the origin (0,0) and are continuous across their domains. Moreover, they both increase as \( x \) increases, indicating a positive correlation between the input and the output.

Differences: The graph of the square root function is defined only for non-negative inputs (x ≥ 0), resulting in a graph that resides in the first quadrant. It increases more slowly for small values of \( x \) and becomes less steep as \( x \) increases. In contrast, the cube root function is defined for all real numbers, exhibiting symmetry about the origin where it extends into both the second and fourth quadrants. It increases steadily but is generally steeper than the square root function near the origin, allowing it to rapidly accommodate negative values, reflecting the fact that \( \sqrt[3]{x} \) can take negative inputs.