Let f(x) = 1 [x]^2 - [x] - 2 + ₁₀(16 - x^2) , where [x] denotes the greatest integer less than or equal to x . If the domain of f(x) is (a, b) [c, d) , then the value of a^2 + b^2 + c^2 + d^2 is equal to :

Step-by-step solution

For f(x) to be defined, two conditions must be satisfied: 1) For the logarithmic term: 16 - x^2 > 0 x^2 x (-4, 4) 2) For the square root in the denominator: [x]^2 - [x] - 2 > 0 ([x] - 2)([x] + 1) > 0 [x] 2 Since [x] is always an integer, the strict inequalities imply: [x] -2 or [x] 3 Using the properties of the greatest integer function: [x] -2 x [x] 3 x 3 So, x (- , -1) [3, ) Taking the intersection of the two conditions: x (-4, 4) ( (- , -1) [3, ) ) x (-4, -1) [3, 4) Comparing this with the given domain (a, b) [c