Which statement about the cube root function y = ∛x is true?

Which statement about the cube root function y = ∛x is true?

• A. It has a constant slope everywhere
• B. It is undefined at x = 0
• C. It is defined for all real x
• D. It has a restricted domain

Answer: (C)

Every real number has a real cube root, so the cube root function is defined for all real x. The graph is an increasing curve, not a straight line, so the slope is not constant across its domain. In fact, as x approaches zero, the slope becomes arbitrarily large in magnitude, giving a vertical tangent at x = 0, even though the function value at 0 is 0. So the statement that it is defined for all real x is true, while the others don’t fit: it isn’t a line with constant slope, it isn’t undefined at x = 0, and there is no restricted domain.

Every real number has a real cube root, so the cube root function is defined for all real x. The graph is an increasing curve, not a straight line, so the slope is not constant across its domain. In fact, as x approaches zero, the slope becomes arbitrarily large in magnitude, giving a vertical tangent at x = 0, even though the function value at 0 is 0. So the statement that it is defined for all real x is true, while the others don’t fit: it isn’t a line with constant slope, it isn’t undefined at x = 0, and there is no restricted domain.