Answer :
The function f(x) = x3 - x is surjective because it takes on all real values, but it is not injective as it fails the horizontal line test, which also means it is not bijective.
The function in question is f(x) = x3 - x. It’s necessary to determine whether this function is injective (one-to-one), surjective (onto), or bijective. To assess if a function is injective, each element of the function's domain should map to a unique element in the codomain, meaning no two distinct elements of the domain map to the same element in the codomain. In this case, the derivative f'(x) = 3x2 - 1, when set to zero, gives us critical points at x = -1/{sqrt}{3} and x = 1/{sqrt}{3}, indicating that the function has a local maximum and minimum, and thus is not injective since it fails the horizontal line test. As for surjectivity, the function needs to cover all elements of the codomain for it to be surjective or onto. Since the cubic function can take on all real values, it is surjective.
Furthermore, for a function to be bijective, it must be both injective and surjective. However, as determined, f(x) is not injective; therefore, it cannot be bijective. In conclusion, f(x) is only surjective and not injective or bijective.
