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

Similarities: Both functions have a domain of all real numbers, but they behave differently in terms of their ranges. The square root function only produces non-negative outputs (range of \( [0, \infty) \)), while the cube root function can produce both positive and negative outputs (range of \( (-\infty, \infty) \)). Additionally, both functions are increasing, meaning that as the input value increases, the output value also increases.

Differences: The shape of the graphs is quite different. The graph of \( f(x) = \sqrt{x} \) starts at the origin and curves upwards, remaining in the first quadrant, whereas the graph of \( g(x) = \sqrt[3]{x} \) passes through the origin and extends into both quadrants, reflecting its ability to take on negative inputs and outputs. This results in a more symmetric appearance around the origin for the cube root function compared to the square root function, which is only in the first quadrant.