Let [ ] denote the greatest integer function. If the domain of the function f(x) = ⁻¹ ( x+[x] 3 ) is [ , ) , then ^2 + ^2 is equal to:

Step-by-step solution

For the function f(x) = ⁻¹ ( x+[x] 3 ) to be defined, the argument of the inverse sine function must lie in the interval [-1, 1] . -1 x+[x] 3 1 -3 x + [x] 3 Using the fractional part function x , we can write x = [x] + x , where 0 x -3 2[x] + x 3 We analyze the possible integer values for [x] : If [x] -2 , then 2[x] + x -4 + x If [x] = -1 , then 2[x] + x = -2 + x . Since 0 x If [x] = 0 , then 2[x] + x = x . Since 0 x If [x] = 1 , then 2[x] + x = 2 + x . Since 0 x If [x] 2 , then 2[x] + x 4 + x 4 > 3 , which does no