Let f : R - 2 R - 1 defined by f (x)= x-3 x-2 and g : R R defined by g (x)=3 x -2 , then sum of all values of x for which f ⁻¹(x)+ g ⁻¹(x)= 19 6 is

Step-by-step solution

Given functions are f(x) = x - 3 x - 2 and g(x) = 3x - 2 . The inverse function f⁻¹(x) is determined by setting y = f(x) and solving for x after interchanging variables: x = y - 3 y - 2 x(y - 2) = y - 3 xy - y = 2x - 3 y(x - 1) = 2x - 3 f⁻¹(x) = 2x - 3 x - 1 , with domain R - 1 . Similarly, g⁻¹(x) yields x = 3y - 2 y = x + 2 3 , so g⁻¹(x) = x + 2 3 , with domain R . Set f⁻¹(x) + g⁻¹(x) = 19 6 : 2x - 3 x - 1 + x + 2 3 = 19 6 . Combining terms with common denominator 3(x - 1) : 3(2x - 3) + (x - 1)(x + 2) 3(x - 1) = 1