Question

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

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.

(4 points)

Answer 1

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

**Similarities:**
Both functions are defined for all real numbers, although their shapes differ significantly. They both pass through the origin (0,0) and are increasing functions: as \( x \) increases, both \( f(x) \) and \( g(x) \) increase in value.

**Differences:**
The most notable difference lies in their rates of growth. The square root function \( f(x) = \sqrt{x} \) grows more slowly as \( x \) increases, while the cube root function \( g(x) = \sqrt[3]{x} \) has a steeper curve for both negative and positive values of \( x \). Furthermore, the square root function is only defined for non-negative \( x \), in contrast to the cube root function which is defined for all \( x \), including negative values. This gives the cube root graph a characteristic shape that dips into the negative quadrant, which the square root function does not exhibit.