Question

Determine which of the following functions are onto:

A. f : R→R defined by f(x)=x³.
B. f : R→R defined by f(x)=x³+x.

Answer 1

Both the functions f(x) = x³ and f(x) = x³ + x are onto, as they are continuous and cover all real number values in their range, either by the nature of cube functions or by being strictly increasing functions respectively.

Step-by-step explanation:

To determine whether the given functions are onto, also known as surjective, we need to check if every element in the codomain (the range for these functions since they map from ℝ to ℝ) has a pre-image in the domain.

f(x) = x³

This function is onto because for every real number y there exists a real number x such that x³ = y. This is because the cube function is continuous and covers all real numbers as x ranges over the real numbers.

f(x) = x³ + x

This function is also onto, as it is continuous and differentiable everywhere, and the derivative f'(x) = 3x² + 1 is always positive, meaning f(x) is strictly increasing and will thus cover all real values as x ranges over the real numbers.