Let g(x) = 1 7 + 3[x-1] - [x]^2 , where [t] denotes the greatest integer less than or equal to t . Then the domain of the function g(x) is

Step-by-step solution

For the function g(x) to be defined, the expression inside the square root must be strictly positive. 7 + 3[x-1] - [x]^2 > 0 Using the property [x-1] = [x] - 1 , we get: 7 + 3([x] - 1) - [x]^2 > 0 7 + 3[x] - 3 - [x]^2 > 0 4 + 3[x] - [x]^2 > 0 [x]^2 - 3[x] - 4 ([x] - 4)([x] + 1) This gives -1 Since [x] must be an integer, the possible values for [x] are 0, 1, 2, 3 . The real values of x that satisfy [x] 0, 1, 2, 3 form the interval [0, 4) . Therefore, the domain of g(x) is [0, 4) . Answer: [0, 4)