What is the domain of the function f(x) = sec^-1(2[x]+1)?

To find the domain of f(x) = ^-1(2[x]+1), we need to consider the domain of the inverse secant function. Step 1: Recall the domain of ^-1(y). The domain of ^-1(y) is y (-, -1] [1, ). This means that the argument of the inverse secant function must be less than or equal to -1, or greater than or equal to 1. Step 2: Apply the domain condition to f(x). In our function, the argument is 2[x]+1. So, we must have: 2[x]+1 -1 or 2[x]+1 1 Step 3: Solve the first inequality for [x]. 2[x]+1 -1 2[x] -1 - 1 2[x] -2 [x] -1 The condition [x] -1 means that the greatest integer less than or equal to x must be -1 or any integer less than -1 (e.g., -2, -3, ). This implies that x must be less than 0. For example, if x = -0.5, [x] = -1. If x = -1.5, [x] = -2. If x = -1, [x] = -1. So, the solution for this inequality is x (-, 0). Step 4: Solve the second inequality for [x]. 2[x]+1 1 2[x] 1 - 1 2[x] 0 [x] 0 The condition [x] 0 means that the greatest integer less than or equal to x must be 0 or any integer greater than 0 (e.g., 1, 2, ). This implies that x must be greater than or equal to 0. For example, if x = 0.5, [x] = 0. If x = 1.5, [x] = 1. If x = 0, [x] = 0. So, the solution for this inequality is x [0, ). Step 5: Combine the solutions from both inequalities. The domain of f(x) is the union of the solutions from Step 3 and Step 4: D = (-, 0) [0, ) This union covers all real numbers. D = (-, ) Comparing this with the given options: (1) (-, -1] [1, ) (2) (-, -1] [0, ) (3) (-, ) (4) (-, ) - \0\ The calculated domain matches option (3). The final answer is (3). 3 done, 2 left today. You're making progress.