Answer :
The function f(x) = x^3 - x is not injective because it is not monotonic over its entire domain and has a local maximum and minimum. However, it is surjective because as a cubic polynomial, it covers all real numbers. So, option (D) is correct.
To determine if the function f(x) = x3 - x is bijective, injective (one-to-one), or surjective (onto), we need to examine its properties. A function is injective if every element of the range is mapped to by at most one element of its domain. A function is surjective if every element of its co-domain is mapped to by at least one element of its domain. Consequently, a function is bijective if it is both injective and surjective.
For the function f(x) = x3 - x, the derivative is f'(x) = 3x2 - 1. If we set the derivative equal to zero to find critical points, we get x = \\pm1/\\sqrt{3}. Since the sign of the derivative changes at these points, the original function has a local maximum and a local minimum, therefore it is not monotonic over its entire domain. As a result, there are distinct real numbers x1 and x2 such that f(x1) = f(x2), which implies that the function is not injective.
However, because the function is a cubic polynomial, as x approaches positive or negative infinity, the function covers all real numbers. Therefore, the function is surjective.
