Answer :
The function f(x) = x³ - x is bijective because it is both injective (one-to-one) and surjective (onto). All options a through e correctly describe the function as it satisfies the properties of being a one-to-one correspondence between all elements of the domain and range. So, option (A) is correct.
The function f(x) = x³ - x is indeed a cubic polynomial, which is odd and differentiable everywhere. To determine whether it is bijective, we need to check two properties: injectivity (one-to-one) and surjectivity (onto).
- Injectivity means each element of the range is mapped by no more than one element of the domain. Since f'(x) = 3x² - 1 can be zero for at most two real numbers, and the cubic polynomial f(x) slopes upward to positive infinity and downward to negative infinity, the function is one-to-one and therefore injective (answer options b and c).
- Surjectivity (onto) means that for every real number y, there exists a real number x such that f(x) = y. This is true for all polynomials of odd degree, so the function is surjective (answer options d and e).
Therefore, the correct answer is that the function f(x) = x³ - x is bijective because it is both injective and surjective. Options a, b, c, d and e, which are synonyms for bijective and its properties, are all correct in this context.
