The domain of the function g(x) = ⁻¹([x - 1]) + _ 0.5 (4 - x^2) , where [t] denotes the greatest integer function, is

Step-by-step solution

For the function g(x) to be defined, both terms must be defined. First, the argument of the inverse cosine must lie in [-1, 1] : -1 [x - 1] 1 -1 x - 1 0 x Thus, x [0, 3) ... (i) Second, the expression inside the square root must be non-negative: _ 0.5 (4 - x^2) 0 Since the base 0.5 is less than 1 , the inequality sign reverses when converting to exponential form. Also, the argument of the logarithm must be strictly positive: 0 0 Solving 4 - x^2 > 0 : x^2 Solving 4 - x^2 1 : x^2 3 x (- , - 3 ] [ 3 , ) ... (iii) Taki