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)

Question

GPT-4o mini · Answer

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.