Determine the domain of the function.

Step 1: Determine the domain of the function. The function f(x) is defined in two parts: • f(x) = 1-x^2 for x 0. This covers all non-positive real numbers. • f(x) = 2x+1 for x > 0. This covers all positive real numbers. Together, these two conditions cover all real numbers. Therefore, the domain of f(x) is (-, ), which is denoted by R. Step 2: Determine the range of the function for x 0. For x 0, f(x) = 1-x^2. This is a parabola opening downwards with its vertex at (0, 1). When x=0, f(0) = 1 - (0)^2 = 1. As x decreases from 0 (e.g., x=-1, -2, ), x^2 increases, so 1-x^2 decreases. For example, f(-1) = 1 - (-1)^2 = 0, f(-2) = 1 - (-2)^2 = -3. Thus, for x 0, the range of f(x) is (-, 1]. Step 3: Determine the range of the function for x > 0. For x > 0, f(x) = 2x+1. This is a linear function with a positive slope. As x approaches 0 from the right, f(x) approaches 2(0)+1 = 1. Since x > 0, the value 1 is not included in this part's range. As x increases from 0, f(x) increases. For example, f(1) = 2(1)+1 = 3, f(2) = 2(2)+1 = 5. Thus, for x > 0, the range of f(x) is (1, ). Step 4: Combine the ranges from both parts. The range of f(x) is the union of the ranges from Step 2 and Step 3: Range = (-, 1] (1, ). This union covers all real numbers. Therefore, the range of f(x) is (-, ), which is denoted by R. The domain is R and the range is R. Comparing this with the given options, option A states "R and R". The final answer is A