Let [t] denote the greatest integer less than or equal to t . If f(x) = ⁻¹ ( 2[x] - 5 7 ) + 1 _e(x^2 - 7x + 13) , then the sum of all integral values of x that lie in the domain of f(x) is :

Step-by-step solution

For the function f(x) to be defined, the following conditions must hold: First, the argument of the inverse sine function must lie in [-1, 1] : -1 2[x] - 5 7 1 -7 2[x] - 5 7 -2 2[x] 12 -1 [x] 6 Since we are looking for integral values of x , for an integer x , [x] = x . Thus, the possible integers satisfying this condition are -1, 0, 1, 2, 3, 4, 5, 6 . Second, the argument of the logarithm must be strictly positive: x^2 - 7x + 13 > 0 The discriminant of this quadratic is = (-7)^2 - 4(1)(13) = 49 - 52 = -3 . Since 0