Consider the function f(x) = ⁻¹(x) + ⁻¹(2x) . If the equation f(x) = k 6 has at least one real solution, then the number of possible integer values of k is :

Step-by-step solution

First, determine the domain of the function f(x) . For ⁻¹(x) to be defined, -1 x 1 . For ⁻¹(2x) to be defined, -1 2x 1 - 1 2 x 1 2 . The intersection of these conditions gives the domain of f(x) as x [- 1 2 , 1 2 ] . Since both ⁻¹(x) and ⁻¹(2x) are strictly increasing functions on their respective domains, their sum f(x) is also strictly increasing on [- 1 2 , 1 2 ] . The minimum value of f(x) occurs at x = - 1 2 : f (- 1 2 ) = ⁻¹ (- 1 2 ) + ⁻¹(-1) = - 6 - 2 = - 2 3 The maximum value of f(x) occurs at x = 1 2 : f (