Let f: R R be a function defined by f(x) = (x + x^2+1 ) + kx , where k R . If f(x) is a bijective function, then the complete set of all possible values of k is :

Step-by-step solution

f(x) = (x + x^2+1 ) + kx f'(x) = 1 x^2+1 + k For f(x) to be bijective, it must be strictly monotonic, meaning f'(x) cannot change sign. Case 1: f'(x) 0 for all x R k - 1 x^2+1 for all x R The maximum value of the right-hand side occurs as x , approaching 0 . Thus, k 0 . Case 2: f'(x) 0 for all x R k - 1 x^2+1 for all x R The minimum value of the right-hand side occurs at x = 0 , which is -1 . Thus, k -1 . For k (- , -1] [0, ) , the linear term kx ensures that the limits as x are or , making the range R (onto). Ther