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

Question

A) Complete an input-output table for the functions f (x) = Vx and
9 (х) = /x. 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 sentences, discuss the similarities and differences between the graphs of the functions.

Answer 1

A) An input-output table for the functions \( f(x) = \sqrt{x} \) (square root) and \( g(x) = \sqrt[3]{x} \) (cube root) would include values such as:

• For \( f(x) = \sqrt{x} \):

  • \( x = 0 \) → \( f(0) = 0 \)
  • \( x = 1 \) → \( f(1) = 1 \)
  • \( x = 4 \) → \( f(4) = 2 \)
  • \( x = 9 \) → \( f(9) = 3 \)
  • \( x = 16 \) → \( f(16) = 4 \)

• For \( g(x) = \sqrt[3]{x} \):

  • \( x = -8 \) → \( g(-8) = -2 \)
  • \( x = 0 \) → \( g(0) = 0 \)
  • \( x = 1 \) → \( g(1) = 1 \)
  • \( x = 8 \) → \( g(8) = 2 \)
  • \( x = 27 \) → \( g(27) = 3 \)

B) The graphs of the functions can be plotted using the points from the input-output table.

C) The graphs of the square root and cube root functions share the origin (0,0) and both increase as \( x \) increases; however, the square root function is only defined for non-negative \( x \), while the cube root function is defined for all real numbers. The square root function exhibits a gradual curve that increases slowly, whereas the cube root function has a steeper slope near the origin and passes through negative values. Overall, while they both represent roots, their domains and growth patterns display significant differences.